The obstacle problem for degenerate doubly nonlinear equations of porous medium type

Abstract

We prove the existence of nonnegative variational solutions to the obstacle problem associated with the degenerate doubly nonlinear equation

$$\begin{aligned} \partial _t b(u) - {{\,\mathrm{div}\,}}(Df(Du)) = 0, \end{aligned}$$

where the nonlinearity \(b :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is increasing, piecewise \(C^1\) and satisfies a polynomial growth condition. The prototype is \(b(u) := u^m\) with \(m \in (0,1)\). Further, \(f :\mathbb {R}^n \rightarrow \mathbb {R}_{\ge 0}\) is convex and fulfills a standard p-growth condition. The proof relies on a nonlinear version of the method of minimizing movements.

Introduction and results

Let \(\Omega \subset \mathbb {R}^n\) be a bounded Lipschitz domain and (0, T) with \(0<T<\infty\) a finite time interval. In the following, \(\Omega _T := \Omega \times (0,T)\) denotes a space-time cylinder. The prototype of the equations considered in the present paper is

$$\begin{aligned} \partial _t u^m - {{\,\mathrm{div}\,}}(|Du|^{p-2}Du) = 0 \quad \text {in } \Omega _T \end{aligned}$$
(1.1)

with parameters \(m \in (0,\infty )\) and \(p \in (1,\infty )\). For \(m=1\) and \(p \in (1,\infty )\), the preceding equation reduces to the parabolic p-Laplace equation, while it is known as the porous medium equation if \(m \in (0,\infty )\) and \(p=2\). Based on the behavior of solutions, doubly nonlinear equations can be subdivided into slow diffusion equations with \(p-1 > m\) and fast diffusion equations with \(p-1 < m\). Further, we distinguish between doubly degenerate equations (\(p>2\), \(0<m<1\)), singular-degenerate equations (\(1<p<2\), \(0<m<1\)), degenerate-singular equations (\(p>2\), \(m>1\)) and doubly singular equations (\(1<p<2\), \(m>1\)), cf. [18]. The porous medium equation and related doubly nonlinear equations are relevant in models for fluid dynamics, filtration and soil science, cf. [4,5,6, 21, 28]. In the present paper, we are concerned with the obstacle problem to doubly nonlinear equations of doubly degenerate and singular-degenerate type. In order to treat (1.1), we use an approach that originates from Lichnewsky and Temam [23] and has later been developed by Bögelein, Duzaar, Marcellini and Scheven [8,9,10,11] to cover a wide range of parabolic problems. More precisely, we are concerned with variational solutions to the Cauchy–Dirichlet problem associated with (1.1) for given initial and boundary values \(g :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\), i.e., functions \(u :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\) satisfying the variational inequality

$$\begin{aligned} \tfrac{1}{p} \iint _{\Omega _T} |Du|^p \,\mathrm {d}x\mathrm {d}t&\le \iint _{\Omega _T} \partial _t v (v^m-u^m) \,\mathrm {d}x\mathrm {d}t+\tfrac{1}{p} \iint _{\Omega _T} |Dv|^p \,\mathrm {d}x\mathrm {d}t\\&\quad -{\mathfrak {B}}[u(T),v(T)] +{\mathfrak {B}}[g(0),v(0)] \end{aligned}$$

associated with (1.2) for any admissible comparison map \(v :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\). Here, we used the abbreviation

$$\begin{aligned} {\mathfrak {B}}[u,v] = \int _\Omega {\mathfrak {b}}[u,v] \,\mathrm {d}x:= \int _\Omega \left[ \tfrac{1}{m+1}v^{m+1} - \tfrac{1}{m+1}u^{m+1} - u^m (v-u)\right] \,\mathrm {d}x\end{aligned}$$

for \(u,v :\Omega \rightarrow \mathbb {R}_{\ge 0}\). Formally, the variational inequality can be derived by multiplying (1.2) by \(v-u\), where \(v :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\) coincides with u on the lateral boundary \(\partial \Omega \times (0,T)\) and then integrating the result over \(\Omega _T\). For the diffusion part, we use integration by parts and the convexity of \(\frac{1}{p}|\cdot |^p\). Finally, by integration by parts the time derivative is shifted from u to v, leading in particular to the integrals over the top and bottom of the space-time cylinder on the right-hand side of the variational inequality. In the present paper, we impose an additional pointwise obstacle condition of the form \(u \ge \psi\) for some obstacle function \(\psi :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\). This means that u is a variational solution to the obstacle problem associated with Eq. (1.2) and initial and boundary values \(g :\Omega \rightarrow \mathbb {R}_{\ge 0}\) if u coincides with g on the parabolic boundary \((\Omega \times \{0\}) \cup (\partial \Omega \times (0,T))\) and satisfies the obstacle condition \(u \ge \psi\) a.e. in \(\Omega _T\) and the preceding variational inequality holds true for any comparison map v with boundary values g and \(v \ge \psi\) a.e. in \(\Omega _T\). For the precise definition, cf. Definition 1.1. At this stage, some words on the history of the problem are in order. First, the seminal work of Grange and Mignot [17] and Alt and Luckhaus [3] should be mentioned. In [3], the authors were among other things concerned with the obstacle problem associated with doubly nonlinear equations of the type

$$\begin{aligned} \partial _t b(u) - {{\,\mathrm{div}\,}}({\mathbf {a}}(b(u),Du)) = {\mathbf {f}}(b(u)), \end{aligned}$$

where b is the gradient of a convex \(C^1\)-function with \(b(0)=0\), \({\mathbf {a}}(b(z),\xi )\) is continuous in z and \(\xi\) and fulfills an ellipticity and \((p-1)\)-growth condition with respect to the gradient variable and \({\mathbf {f}}(b(z))\) is continuous in z and satisfies a suitable growth condition. Further, a two-sided obstacle condition is imposed with obstacle functions \(\psi _\pm \in L^p(0,T;W^{1,p}(\Omega )) \cap L^\infty (\Omega _T)\) with \(\partial _t \psi _\pm \in L^1(\Omega _T)\) and \(\psi _- \le \psi _+\) a.e. in \(\Omega _T\). Under these assumptions, the existence of variational solutions has been established via time discretization for Neumann boundary values. However, the proof extends to the case of an additional Dirichlet boundary condition with zero boundary values on a part of the boundary. Later, Bernis [7] showed the existence of weak solutions to the Cauchy problem associated with higher order doubly nonlinear equations on unbounded domains. Further, Ivanov, Mkrtychyan and Jäger [18,19,20] used regularization and a priori Hölder estimates to prove the existence of regular weak solutions to the Cauchy–Dirichlet problem associated with doubly nonlinear equations. The boundary values satisfy \(g \in W^{1,p}(\Omega _T) \cap L^\infty (\Omega _T)\) and an additional continuity assumption with respect to space and time. A different approach has been pursued by Akagi and Stefanelli [2] in order to treat the Cauchy–Dirichlet problem with homogenous Dirichlet boundary values associated with doubly nonlinear equations of the type

$$\begin{aligned} \partial _t b(u) - {{\,\mathrm{div}\,}}({\mathbf {a}}(Du)) \ni {\mathbf {f}}, \end{aligned}$$

where \(b \subset \mathbb {R}\times \mathbb {R}\) and \({\mathbf {a}} \subset \mathbb {R}^n \times \mathbb {R}^n\) are maximal monotone graphs that fulfill polynomial growth conditions. The authors solve the problem by means of elliptic regularization (Weighted Energy Dissipation Functional method) after transforming it into the dual formulation \(-{{\,\mathrm{div}\,}}({\mathbf {a}}(Db^{-1}(v)) \ni {\mathbf {f}} - \partial _t v\). Recently, by a nonlinear version of the method of minimizing movements Bögelein, Duzaar, Marcellini and Scheven [11] were able to prove the existence of nonnegative variational solutions to the Cauchy–Dirichlet problem with time-independent boundary values associated with

$$\begin{aligned} \partial _t b(u) - {{\,\mathrm{div}\,}}(D_\xi f(x,u,Du)) = -D_uf(x,u,Du), \end{aligned}$$

where \(b :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is continuous, piecewise \(C^1\) and satisfies a polynomial growth condition with \(b(0)=0\). Further, \((u,\xi ) \mapsto f(x,u,\xi )\) is convex for a.e. \(x \in \Omega\) and f satisfies a coercivity, but not necessarily a growth condition. This allows f to have nonstandard growth like exponential or (pq)-growth with \(1<p<q<\infty\). Note that the required nonnegativity of the solutions is an obstacle condition with obstacle function \(\psi \equiv 0\). In the case of singly nonlinear equations of p-Laplace type, Bögelein, Duzaar and Scheven [12] established the existence of variational solutions to the obstacle problem with a far more general obstacle function \(\psi \in L^2(\Omega _T) \cap L^p(0,T;W^{1,p}(\Omega ))\) and time-dependent boundary values via the classical method of minimizing movements. Finally, the author has been concerned with the singular equation/system

$$\begin{aligned} \partial _t \big (|u|^{m-1}u\big ) - {{\,\mathrm{div}\,}}(Df(Du)) = 0, \end{aligned}$$

where \(m>1\), f is convex and satisfies a standard p-growth and coercivity condition. By the nonlinear minimizing movements scheme developed in [11] and suitable approximation arguments, the existence of signed or vector-valued variational solutions to the Cauchy–Dirichlet problem with time-dependent boundary values and the existence of solutions to the obstacle problem with time-dependent obstacle function have been established, cf. [26, 27]. More precisely, the boundary values and the obstacle function are contained in the space \(L^p(0,T;W^{1,p}(\Omega ))\) with time derivative in \(L^1(0,T;L^{m+1}(\Omega ))\) and initial values in \(L^{m+1}(\Omega )\). In the present paper, the question of uniqueness will not be discussed, since this is a delicate and widely open issue for doubly nonlinear equations. We refer to [15] for a counterexample and to [3, 15] for sufficient conditions.

The general doubly nonlinear equation

In the present paper, we are concerned with the doubly nonlinear equation

$$\begin{aligned} \partial _t b(u) - {{\,\mathrm{div}\,}}(D_\xi f(Du)) = 0 \quad \text {in } \Omega _T. \end{aligned}$$
(1.2)

Here, we assume that \(f :\mathbb {R}^n \rightarrow \mathbb {R}_{\ge 0}\) is a Borel-measurable, convex function that fulfills the growth and coercivity condition

$$\begin{aligned} \nu |\xi |^p \le f(\xi ) \le L\big (1+|\xi |^p\big ) \end{aligned}$$
(1.3)

with constants \(0<\nu \le L\) for all \(\xi \in \mathbb {R}^n\). Observe that (1.3) and the convexity of f together imply that f is locally Lipschitz continuous. More precisely,

$$\begin{aligned} |f(\xi )-f(\eta )| \le c(n,p,L)\big (1+|\xi |^{p-1}+|\eta |^{p-1}\big )|\xi -\eta | \end{aligned}$$
(1.4)

holds true for any \(\xi ,\eta \in \mathbb {R}^n\), cf. [24, Eq. (2.9)]. Further, the nonlinearity \(b :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is continuous and piecewise \(C^1\) in \(\mathbb {R}_{>0}\). Replacing b(u) by \(b(u)-b(0)\), we suppose without loss of generality that \(b(0)=0\). Moreover, we assume that there exist constants \(0<\ell \le m \le 1\) such that

$$\begin{aligned} \ell \le \frac{ub'(u)}{b(u)} \le m \end{aligned}$$
(1.5)

holds true whenever \(u>0\), \(b(u)>0\) and \(b'(u)\) exists. In particular, this implies that \(b'(u) \ge 0\) if it exists. Then, the primitive of b defined by

$$\begin{aligned} \Phi (u) := \int _0^u b(s) \,\mathrm {d}s\quad \quad \text {for any } u \ge 0 \end{aligned}$$

is a convex \(C^1\) function with \(\Phi (0)=0\). Further, the convex conjugate (Fenchel conjugate) of \(\Phi\) is defined by

$$\begin{aligned} \Phi ^*(v) := \sup _{u \ge 0} \{uv - \Phi (u)\} \quad \quad \text {for any } v \ge 0, \end{aligned}$$

which immediately implies Fenchel’s inequality

$$\begin{aligned} uv \le \Phi (u) + \Phi ^*(v) \quad \quad \text {for all } u,v \ge 0. \end{aligned}$$

Since \(\Phi\) is convex, we easily compute that equality holds for \(v=b(u)\), i.e.

$$\begin{aligned} \Phi ^*(b(u)) = b(u)u - \Phi (u) \quad \quad \text {for any } u \ge 0. \end{aligned}$$

At this stage, we define

$$\begin{aligned} {\mathfrak {b}}[u,v]&:= \Phi (v) - \Phi (u) - b(u)(v-u) \\&= \Phi (v) + \Phi ^*(b(u)) - b(u)v \end{aligned}$$

for any \(u,v \ge 0\). In the variational inequality associated with (1.2), we will use boundary terms

$$\begin{aligned} {\mathfrak {B}}[u,v] := \int _\Omega {\mathfrak {b}}[u,v] \,\mathrm {d}x\end{aligned}$$

for functions \(u,v :\Omega \rightarrow \mathbb {R}_{\ge 0}\). Furthermore, in order to be able to formulate solutions to the obstacle problem, we define the Orlicz space related to \(\Phi\) and some domain \(A \subset \mathbb {R}^k\), \(k \in \mathbb {N}\), by

$$\begin{aligned} L^\Phi (A) := \left\{ v :A \rightarrow \mathbb {R}\;\text {measurable } :\int _A \Phi (a|v|) \,\mathrm {d}x< \infty \quad \text {for some } a>0 \right\} . \end{aligned}$$

For details on Orlicz spaces, we refer to the monographs [1, 25]. By the assumptions on b, we obtain that both \(\Phi\) and \(\Phi ^*\) satisfy the \(\nabla _2\) and the \(\Delta _2\) condition (see (2.3)). In particular, the \(\Delta _2\) condition on \(\Phi\) implies that an equivalent definition of the Orlicz space above is given by

$$\begin{aligned} L^\Phi (A) := \left\{ v :A \rightarrow \mathbb {R}\;\text {measurable } :\int _A \Phi (|v|) \,\mathrm {d}x< \infty \right\} . \end{aligned}$$

Henceforth, we often abbreviate the modular (see [25, Chapter III.3.4]) by

$$\begin{aligned} \varrho _A(v) := \int _A \Phi (|v|) \,\mathrm {d}x. \end{aligned}$$

In the present paper, we assume that \(L^\Phi (A)\) is equipped with the Orlicz norm

$$\begin{aligned} \Vert v\Vert _{L^\Phi (A)} := \sup \left\{ \bigg |\int _A vw \,\mathrm {d}x\bigg | :\int _A \Phi ^*(|w|) \,\mathrm {d}x\le 1\right\} , \end{aligned}$$

which is equivalent to the Luxemburg norm

$$\begin{aligned} \Vert v\Vert '_{L^{\Phi }(A)} := \inf \left\{ \lambda >0 :\int _A \Phi \left( \tfrac{|v|}{\lambda }\right) \,\mathrm {d}x\le 1 \right\} . \end{aligned}$$

Dealing with these norms is not always straightforward. However, since \(\Phi\) fulfills the \(\Delta _2\) condition, norm convergence is equivalent to modular convergence, i.e.,

$$\begin{aligned} v_i \rightarrow v\, \text{strongly in}\,\, L^\Phi (A)\, \text{ as }i \rightarrow \infty \quad \Leftrightarrow \quad \lim _{i \rightarrow \infty } \varrho _A(v_i-v) = 0 \end{aligned}$$
(1.6)

holds true for any \(v_i, v \in L^\Phi (A)\), \(i \in \mathbb {N}\), and for sets \(S \subset L^\Phi (A)\) we know that

$$\begin{aligned} S \text{ is } \text{ norm } \text{ bounded } \quad \Leftrightarrow \quad \sup _{v \in S} \varrho _A(v) < \infty , \end{aligned}$$
(1.7)

cf. [25, Chapter III.3.4]. Analogously, we define the Orlicz space \(L^{\Phi ^*}(A)\), the modular \(\varrho _A^*(\cdot )\) and the norms \(\Vert \cdot \Vert _{L^{\Phi ^*}(A)}\) and \(\Vert \cdot \Vert '_{L^{\Phi ^*}(A)}\) related to the convex conjugate \(\Phi ^*\). In this setting, the generalized Hölder’s inequality

$$\begin{aligned} \int _A |vw| \,\mathrm {d}x\le \Vert v\Vert _{L^\Phi (A)} \Vert w\Vert '_{L^{\Phi ^*}(A)} \end{aligned}$$
(1.8)

holds true for functions \(v \in L^\Phi (A)\), \(w \in L^{\Phi ^*}(A)\), cf. [25, Chapter III.3.3]. Further, the \(\Delta _2\) condition on \(\Phi\) implies that \(L^\Phi (\Omega )\) is separable and that the dual space of \((L^\Phi (A),\Vert \cdot \Vert _{L^\Phi (A)})\) is isometrically isomorphic to \((L^{\Phi ^*}(A),\Vert \cdot \Vert '_{L^{\Phi ^*}(A)})\), cf. [25, Chapters III, IV].

The main result

In order to formulate a boundary condition, we consider the affine parabolic space \(g + L^p(0,T;W^{1,p}_0(\Omega ))\) consisting of the functions \(v \in L^p(0,T;W^{1,p}(\Omega ))\) such that \(v(t) \in g(t) + W^{1,p}_0(\Omega )\) for a.e. \(t \in (0,T)\). In the present paper, we assume that nonnegative boundary values \(g :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\) are given by

$$\begin{aligned} g \in L^p(0,T;W^{1,p}(\Omega )) \text{ with } \partial _t g \in L^1(0,T;L^\Phi (\Omega )) \text{ and } g_o :=g(0) \in L^\Phi (\Omega ) \end{aligned}$$
(1.9)

and that the nonnegative obstacle function \(\psi :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\) satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} \psi \in g+ L^p(0,T;W^{1,p}_0(\Omega )) \text{ with } \partial _t \psi \in L^1(0,T;L^\Phi (\Omega )), \psi (0) \in L^\Phi (\Omega ) \\ \text{ and } \psi \le g \text{ a.e. } \text{ in } \Omega _T. \end{array} \right. \end{aligned}$$
(1.10)

Definition 1.1

(Variational solution) Assume the convex integrand f satisfies (1.3) and that (1.9) and (1.10) hold true. A measurable nonnegative map \(u :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\) in the class

$$\begin{aligned} u \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}_0(\Omega ))\big ) \text{ with } u \ge \psi \text{ a.e. } \text{ in } \Omega _T \end{aligned}$$

is called a variational solution to the obstacle problem associated with (1.2) if and only if it solves the variational inequality

$$\begin{aligned} \iint _{\Omega _\tau } f(Du) \,\mathrm {d}x\mathrm {d}t&\le \iint _{\Omega _\tau } \partial _t v (b(v)-b(u)) \,\mathrm {d}x\mathrm {d}t+\iint _{\Omega _\tau } f(Dv) \,\mathrm {d}x\mathrm {d}t\nonumber \\&\quad -{\mathfrak {B}}[u(\tau ),v(\tau )] +{\mathfrak {B}}[g_o,v(0)] \end{aligned}$$
(1.11)

for a.e. \(\tau \in [0,T]\) and any comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\).

At this stage, we are able to state the main result of the present paper. Note that we can conclude from (1.11) that u attains the initial datum \(g_o\) in the \(L^\Phi\)-sense; see Lemma 2.18.

Theorem 1.2

Assume that the convex integrand f fulfills (1.3) and that the hypotheses (1.9) and (1.10) are satisfied. Then, there exists a variational solution

$$\begin{aligned} u \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}_0(\Omega ))\big ) \text{ with } u \ge \psi \text{ a.e. } \text{ in } \Omega _T \end{aligned}$$

to (1.2) in the sense of Definition1.1. Furthermore, u attains the initial datum \(g_o\) in the \(L^\Phi\)-sense.

Methods of proof

First, in Sect. 2, we collect lemmas that we need in the subsequent proofs of the existence theorems. Their proofs are already known or easy. Next, in Sect. 3, we prove a preliminary existence result for regular data, i.e., boundary values and an obstacle with time derivative in \(L^2(\Omega _T) \cap L^p(0,T;W^{1,p}(\Omega ))\) and initial values in \(L^2(\Omega ) \cap W^{1,p}(\Omega )\). Since \(b'(0)\) is infinite, we assume that g and \(\psi\) are bounded away from zero. The proof relies on a nonlinear version of the method of minimizing movements. More precisely, we fix a step size \(h_K :=T/K\) for some \(K \in \mathbb {N}\) and consider time slices of \(\Omega _T\) at the time points \(ih_K\), \(i \in \{0,\ldots ,K\}\). Then, we set \(u_0 = g(0)\) and iteratively define minimizers \(u_i\) of the elliptic variational functionals

$$\begin{aligned} F_i[v] := \int _\Omega f(Dv) \,\mathrm {d}x+\tfrac{1}{h} \int _\Omega {\mathfrak {b}}[u_{i-1},v] \,\mathrm {d}x\end{aligned}$$

in the class \(v \in L^\Phi (\Omega ) \cap (g(ih_K) + W^{1,p}_0(\Omega ))\). Observe that \({\mathfrak {b}}[u,v] = \tfrac{1}{2} \Vert u-v\Vert _{L^2(\Omega )}^2\) if \(b(u)=u\) and hence the scheme reduces to the classical method of minimizing movements in the linear case. Next, in Sect. 3.2, we derive suitable energy estimates for the minimizers \(u_i\). Here, the stronger assumptions on the data are crucial. As in the classical scheme, we assemble the functions \(u_i\) to a map \(u^{(K)} :\Omega \times (-h_K,T] \rightarrow \mathbb {R}_{\ge 0}\) that is piecewise constant with respect to time by setting \(u^{(K)}(t) := u_i\) for \(t \in ((i-1)h_K,ih_K]\), \(i \in \{0,\ldots ,K\}\). By the energy estimates from Sect. 3.2 and the compactness result 2.21, we find a subsequence and a suitable limit map \(u \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}_0(\Omega ))\big )\) such that \(u^{(K)} \rightharpoondown u\) weakly in \(L^p(0,T;W^{1,p}(\Omega ))\) and \(u^{(K)} \rightarrow u\) a.e. in \(\Omega _T\). In Sect. 3.4, we assemble the functionals \(F_i\) such that \(u^{(K)}\) inherits a minimizing property and thus deduce a preliminary variational inequality for \(u^{(K)}\). Finally, in Sect. 3.5, we pass to the limit \(K \rightarrow \infty\) in these preliminary inequalities, which allows us to show that u is the desired variational solution. In Sect. 4, we relax the regularity assumptions on the spatial variables of the data. More precisely, the time derivatives of the boundary values and obstacle are now contained in \(L^2(0,T;L^\Phi (\Omega )) \cap L^p(0,T;W^{1,p}(\Omega ))\) and the initial values in \(L^\Phi (\Omega )\). Further, g and \(\psi\) may attain the value zero. The proof of the existence result relies on standard mollification of the boundary values and obstacle with respect to the spatial variables. Since the regularized data \(g_\varepsilon\) and \(\psi _\varepsilon\) satisfy the assumptions of Sect. 3, we find variational solutions \(u_\varepsilon\), \(\varepsilon >0\), corresponding to \(g_\varepsilon\) and \(\psi _\varepsilon\). By the energy bound from Lemma 2.20, we deduce that a subsequence converges weakly to a suitable limit map \(u \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}_0(\Omega ))\big )\). Passing to the limit \(\varepsilon \downarrow 0\) in the variational inequalities fulfilled by \(u_\varepsilon\), we conclude that u is the desired variational inequality to g and \(\psi\). To this end, it is important to understand that weak convergence \(u_\varepsilon \mathop {\rightharpoondown }\limits ^{*}u\) weakly\(^*\) in \(L^\infty (0,T;L^\Phi (\Omega ))\) as \(\varepsilon \downarrow 0\) in general does not imply \(b(u_\varepsilon ) \mathop {\rightharpoondown }\limits ^{*}b(u)\) weakly\(^*\) in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\) as \(\varepsilon \downarrow 0\). Even if there is a convergent subsequence, the limit might not be b(u). Therefore, we need to use a technique similar to the one in [13, Lemma 9.1] to establish the desired convergence assertion. Finally, in Sect. 5, we give the proof of Theorem 1.2. The technique is similar to the one in Sect. 4, but based on the time mollification procedure described in Sect. 2.3 instead of standard mollification.

Preliminaries

Technical lemmas

In this section, we collect some lemmas that we will need for the proof of the existence result. For the proofs of the lemmas 2.1, 2.3, 2.6, 2.7, 2.9 and 2.10, we refer to [11, Section 2.1].

Lemma 2.1

For any continuous, piecewise \(C^1\) function \(b :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) satisfying (1.5) and any \(\lambda >1\), \(u>0\) , we have that:

$$\begin{aligned} \lambda ^\ell b(u)&\le b(\lambda u) \le \lambda ^m b(u), \end{aligned}$$
(2.1)
$$\begin{aligned} \tfrac{\ell }{m}\lambda ^{\ell -1} b'(u)&\le b'(\lambda u) \le \tfrac{m}{\ell } \lambda ^{m-1} b'(u), \end{aligned}$$
(2.2)
$$\begin{aligned} \lambda ^{\ell +1} \Phi (u)&\le \Phi (\lambda u) \le \lambda ^{m+1} \Phi (u), \end{aligned}$$
(2.3)
$$\begin{aligned} \lambda ^\frac{m+1}{m} \Phi ^*(u)&\le \Phi ^*(\lambda u) \le \lambda ^\frac{\ell +1}{\ell } \Phi ^*(u). \end{aligned}$$
(2.4)

Remark 2.2

Assuming (1.5), we infer from Lemma 2.1 that

$$\begin{aligned} b(1) \min \{u^\ell ,u^m\} \le b(u) \le b(1) \max \{u^\ell ,u^m\} \end{aligned}$$

for any \(u>0\).

Lemma 2.3

Assume that b satisfies (1.5). Then,

$$\begin{aligned} \tfrac{1}{m+1} u b(u) \le \Phi (u) \le \tfrac{1}{\ell } \Phi ^*(b(u)) \le \tfrac{m}{\ell (m+1)} u b(u) \end{aligned}$$

holds true for any \(u \ge 0\).

Remark 2.4

Combining Lemma 2.3 with (1.7), we find that \(v \in L^\infty (0,T;L^\Phi (\Omega ))\) implies \(b(v) \in L^\infty (0,T;L^{\Phi ^*}(\Omega ))\).

Lemma 2.5

For any \(v \in L^\Phi (\Omega )\) , we have that

$$\begin{aligned} \Vert v\Vert '_{L^{\Phi }(\Omega )} \le 1 + \varrho _\Omega (v)^\frac{1}{\ell +1}. \end{aligned}$$
(2.5)

Proof

Set

$$\begin{aligned} M := \varrho _\Omega (v) < \infty . \end{aligned}$$

If \(M \le 1\), by definition of the Luxemburg norm, we have \(\Vert v\Vert '_{L^{\Phi }(\Omega )} \le 1\). On the other hand, if \(M>1\), by Lemma 2.1, we find that

$$\begin{aligned} 1 \ge \tfrac{1}{M} \varrho _\Omega (v) \ge \varrho _\Omega (M^{-\frac{1}{\ell +1}}|v|) \end{aligned}$$

and therefore that \(\Vert v\Vert '_{L^{\Phi }(\Omega )} \le M^\frac{1}{\ell +1}\). Combining the cases implies (2.5). \(\square\)

Lemma 2.6

Assume that b satisfies (1.5). Then, for all \(u, v \ge 0\), we have that

$$\begin{aligned} \Phi ^*(b(u))&\le 2 {\mathfrak {b}}[u,v] + 2^{m+2} \Phi (v), \\ \Phi (v)&\le 2 {\mathfrak {b}}[u,v] + 2^{2+\tfrac{1}{\ell }} \Phi ^*(b(u)). \end{aligned}$$

Lemma 2.7

Assume that (1.5) is satisfied. Then, there exists a constant \(c=c(m,\ell )\) such that the estimates

$$\begin{aligned} {\mathfrak {b}}[u,v]&\le (b(v)-b(u))(v-u) \\&\le \big |\sqrt{vb(v)}-\sqrt{ub(u)}\big |^2 \\&\le c \, \big |\sqrt{\Phi (v)}-\sqrt{\Phi (u)}\big |^2 \\&\le c^2 \, {\mathfrak {b}}[u,v] \end{aligned}$$

hold true for all \(u,v \ge 0\).

Lemma 2.8

Assume that (1.5) is in force. If \((v_i)_{i \in \mathbb {N}}\) is a sequence in \(L^\Phi (\Omega )\) and \(v \in L^\Phi (\Omega )\) such that \(v_i \rightarrow v\) in \(L^\Phi (\Omega )\) as \(i \rightarrow \infty\), we also have that \(v_i \rightarrow v\) in \(L^{\ell +1}(\Omega )\).

Proof

The Lebesgue space \(L^{\ell +1}(\Omega )\) is obviously related to the function \(\Psi (x) := |x|^{\ell +1}\). By (2.3) for any \(\varepsilon >0\), we obtain that

$$\begin{aligned} \left| \tfrac{x}{\varepsilon }\right| ^{\ell +1} = \varepsilon ^{-\ell -1}\Phi (1)^{-1} x^{\ell +1}\Phi (1) \le \varepsilon ^{-\ell -1}\Phi (1)^{-1} \Phi (x) \qquad \forall x \ge 1. \end{aligned}$$

Hence, \(\Phi\) is completely stronger than \(\Psi\), cf. [25, Definition 2.2.1]. By (1.6), we find that \(\lim _{i \rightarrow \infty } \varrho _\Omega (v_i-v) = 0\). Therefore, from the definition of the Luxemburg norm and [25, Theorem 5.3.1], we infer

$$\begin{aligned} \lim _{i \rightarrow \infty } \Vert v_i-v\Vert _{L^{\ell +1}(\Omega )} = \lim _{i \rightarrow \infty } \Vert v_i-v\Vert '_{L^{\Psi }(\Omega )} =0. \end{aligned}$$

This concludes the proof of the lemma. \(\square\)

Lemma 2.9

Assume that (1.5) holds true. If \((u_i)_{i \in \mathbb {N}}\) is a sequence in \(L^\Phi (\Omega )\) such that \(\Phi (u_i) \rightarrow \Phi (u)\) in \(L^1(\Omega )\) for some \(u \in L^\Phi (\Omega )\), , then \(u_i \rightarrow u\) in \(L^\Phi (\Omega )\) as \(i \rightarrow \infty\). In particular, we have that

$$\begin{aligned} \varrho _\Omega (u_i-u_o) \le \int _\Omega |\Phi (u_i) - \Phi (u_o)| \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

Lemma 2.10

Suppose that b satisfies (1.5). If \(v \in L^1(\Omega _T)\) is given with \(\partial _t v \in L^\Phi (\Omega _T)\) and \(v(0) \in L^\Phi (\Omega )\), , we have that \(v \in C^0([0,T];L^\Phi (\Omega ))\).

Lemma 2.11

Assume that the functions \(v,\psi \in C^0([0,T];L^\Phi (\Omega ))\) satisfy \(v \ge \psi\) a.e. in \(\Omega _T\) with respect to the \((n+1)\)-dimensional Lebesgue measure \(\mathcal {L}^{n+1}\). Then, \(v(0) \ge \psi (0)\) a.e. in \(\Omega\) with respect to the n-dimensional Lebesgue measure \(\mathcal {L}^n\).

Difference quotients

First, adapting the proof of [16, Theorem 1.33], we show the following variant of Lebesgue’s differentiation theorem.

Lemma 2.12

Let \((X,\Vert \cdot \Vert _X)\) be a separable Banach space and \(v \in L^1(0,T;X)\). Then, for a.e. \(t \in [0,T]\) , we have that

$$\begin{aligned} \lim _{h \downarrow 0} \mathop {\int \!\!\!\!\!\!-}\nolimits _t^{t+h} \Vert v(s)-v(t)\Vert _X \,\mathrm {d}s=0. \end{aligned}$$

Proof

Since X is separable, there exists a dense subset \((v_i)_{i \in \mathbb {N}} \subset X\). Then, we know that \(t \mapsto \Vert v(t)-v_i\Vert _X \in L^1(0,T)\) for any \(i \in \mathbb {N}\). By Lebesgue’s differentiation theorem, we conclude that

$$\begin{aligned} \lim _{h \downarrow 0} \mathop {\int \!\!\!\!\!\!-}\nolimits _t^{t+h} \Vert v(s)-v_i\Vert _X \,\mathrm {d}s= \Vert v(t)-v_i\Vert _X \end{aligned}$$

holds true for any \(t \in [0,T] \setminus \mathcal {N}_i\), where \(\mathcal {N}_i\), \(i \in \mathbb {N}\), denotes a \(\mathcal {L}^1\)-null set. Consequently, \(\mathcal {N} := \bigcup _{i \in \mathbb {N}} \mathcal {N}_i \cup \{t \in [0,T] :v(t) \notin X\}\) is a \(\mathcal {L}^1\)-null set and

$$\begin{aligned} \lim _{h \downarrow 0} \mathop {\int \!\!\!\!\!\!-}\nolimits _t^{t+h} \Vert v(s)-v_i\Vert _X \,\mathrm {d}s= \Vert v(t)-v_i\Vert _X \end{aligned}$$

holds true for any \(t \in [0,T] \setminus \mathcal {N}\) and any \(i \in \mathbb {N}\). Next, fix \(t \in [0,T] \setminus \mathcal {N}\) and let \(\varepsilon >0\). Since \((v_i)_{i \in \mathbb {N}}\) is dense in X, there exists \(i \in \mathbb {N}\) such that

$$\begin{aligned} \Vert v(t)-v_i\Vert _X < \tfrac{\varepsilon }{2}. \end{aligned}$$

Combining the preceding considerations, we infer

$$\begin{aligned} \limsup _{h \downarrow 0} \mathop {\int \!\!\!\!\!\!-}\nolimits _t^{t+h} \Vert v(s)-v(t)\Vert _X \,\mathrm {d}s&\le \lim _{h \downarrow 0} \mathop {\int \!\!\!\!\!\!-}\nolimits _t^{t+h} \Vert v(s)-v_i\Vert _X \,\mathrm {d}s+\Vert v_i-v(t)\Vert _X \\&= 2\Vert v_i-v(t)\Vert _X <\varepsilon . \end{aligned}$$

Since \(\varepsilon >0\) was arbitrary, this yields the claim. \(\square\)

Let \(h>0\). The difference quotient of a function v with respect to time is denoted by

$$\begin{aligned} \Delta _h v (t) := \tfrac{1}{h} \big (v(t+h)-v(t)\big ). \end{aligned}$$

We prove the following convergence assertion for \(X=L^\Phi (\Omega )\).

Lemma 2.13

Assume that \(v \in C^0([0,T];L^\Phi (\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\). Further, let \(h_k :=T/k\) for some \(k \in \mathbb {N}\) and define the piecewise constant function \(v^{(k)} :\Omega _T \rightarrow \mathbb {R}\) by

$$\begin{aligned} v^{(k)} := v(ih_k) \text{ for } t \in ((i-1)h_k,ih_k], i \in \{1,\ldots ,k\}. \end{aligned}$$

Then, we have that

$$\begin{aligned} \Delta _{h_k} v^{(k)} \rightarrow \partial _t v \text{ in } L^1(0,T;L^\Phi (\Omega )) \text{ as } k \rightarrow \infty \text{. } \end{aligned}$$

Proof

Fix \(t \in [0,T]\) and let \(i \in \{1,\ldots ,k\}\) such that \(t \in ((i-1)h_k,ih_k]\). Observe that

$$\begin{aligned} \Delta _{h_k} v^{(k)}(t) = \tfrac{1}{h_k} \big (v((i+1)h_k) - v(ih_k)\big ) = \mathop {\int \!\!\!\!\!\!-}\nolimits _{ih_k}^{(i+1)h_k} \partial _t v(s) \,\mathrm {d}s. \end{aligned}$$

Therefore, choosing suitable \(i \in \{1,\ldots ,k\}\) and applying Lemma 2.12 with \(X=L^\Phi (\Omega )\) we compute that

$$\begin{aligned} \Vert \Delta _{h_k} v^{(k)}(t) - \partial _t v(t)\Vert _{L^\Phi (\Omega )}&= \bigg \Vert \mathop {\int \!\!\!\!\!\!-}\nolimits _{ih_k}^{(i+1)h_k} \big [\partial _t v(s) - \partial _t v(t)\big ] \,\mathrm {d}s\bigg \Vert _{L^\Phi (\Omega )} \\&\le \mathop {\int \!\!\!\!\!\!-}\nolimits _{ih_k}^{(i+1)h_k} \Vert \partial _t v(s) - \partial _t v(t)\Vert _{L^\Phi (\Omega )} \,\mathrm {d}s\\&\le 2\mathop {\int \!\!\!\!\!\!-}\nolimits _{t}^{t+2h_k} \Vert \partial _t v(s) - \partial _t v(t)\Vert _{L^\Phi (\Omega )} \,\mathrm {d}s\rightarrow 0 \end{aligned}$$

a.e. in [0, T] as \(k \rightarrow \infty\). Moreover, in a similar way, we find that

$$\begin{aligned} \Vert \Delta _{h_k} v^{(k)}(t) - \partial _t v(t)\Vert _{L^\Phi (\Omega )}&\le 2\mathop {\int \!\!\!\!\!\!-}\nolimits _{t}^{t+2h_k} \Vert \partial _t v(s)\Vert _{L^\Phi (\Omega )} \,\mathrm {d}s+\Vert \partial _t v\Vert _{L^\Phi (\Omega )}\\&\rightarrow 3\Vert \partial _t v\Vert _{L^\Phi (\Omega )} \end{aligned}$$

in \(L^1(0,T)\) in the limit \(k \rightarrow \infty\). Here, we used that \(\mathop {\int \!\!\!\!\!\!-}\nolimits _{t}^{t+2h_k} \Vert \partial _t v(s)\Vert _{L^\Phi (\Omega )} \,\mathrm {d}s\) is the Steklov average of the function \(t \mapsto \Vert \partial _t v(t)\Vert _{L^\Phi (\Omega )} \in L^1(0,T)\). Hence, a version of the dominated convergence theorem (cf. [16, Theorem 1.20]) implies the claim. \(\square\)

The following statement is a slightly different version of the discrete integration by parts formula [11, Lemma 2.10].

Lemma 2.14

Let \(h \in (0,1]\), and \(u,v \in L^\Phi (\Omega \times (-h,T+h))\) be two nonnegative functions. Then, the following integration by parts formula

$$\begin{aligned} \iint _{\Omega _T} \Delta _{-h} b(u) (v-u) \,\mathrm {d}x\mathrm {d}t&\le \iint _{\Omega _T} \Delta _h v (b(v)-b(u)) \,\mathrm {d}x\mathrm {d}t\\&\quad -\tfrac{1}{h} \iint _{\Omega \times (T-h,T)} {\mathfrak {b}}[u(t),v(t+h)] \,\mathrm {d}x\mathrm {d}t\\&\quad +\tfrac{1}{h} \iint _{\Omega \times (-h,0)} {\mathfrak {b}}[u,v] \,\mathrm {d}x\mathrm {d}t+ \varvec{\delta }_1(h) + \varvec{\delta }_2(h), \end{aligned}$$

holds true, where the error terms \(\varvec{\delta }_1(h)\) and \(\varvec{\delta }_2(h)\) are given by

$$\begin{aligned} \varvec{\delta }_1(h)&:= \tfrac{1}{h} \iint _{\Omega _T} {\mathfrak {b}}[v(t),v(t+h)] \,\mathrm {d}x\mathrm {d}t, \\ \varvec{\delta }_2(h)&:= \iint _{\Omega \times (-h,0)} \Delta _h v (b(v(t+h))-b(u(t))) \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

If we assume additionally that \(v \in L^\infty (-h_o,T+h_o;L^\Phi (\Omega ))\) and \(\partial _t v \in L^1(-h_o,T+h_o;L^\Phi (\Omega ))\) for some \(h_o>0\), then we have

$$\begin{aligned} \lim _{h \downarrow 0} \varvec{\delta }_1(h) = 0 \quad \text {and}\quad \lim _{h \downarrow 0} \varvec{\delta }_2(h) = 0. \end{aligned}$$

Proof

For the proof of the integration by parts formula, we refer to [11, Lemma 2.10]. It remains to show the second assertion of the lemma. By Lemma 2.7, we conclude that

$$\begin{aligned} 0 \le \tfrac{1}{h} {\mathfrak {b}}\big [v(t),v(t+h)\big ] \le \big (b(v(t+h))-b(v(t))\big ) \Delta _h v(t). \end{aligned}$$

First, observe that \(\big (b(v(t+h))-b(v(t))\big ) \Delta _h v(t) \rightarrow 0\) a.e. in \(\Omega _T\) as \(h \downarrow 0\). Further, we have that

$$\begin{aligned} \sup _{h \in (0,h_o)} \sup _{t \in [0,T]} \Vert b(v(t+h))-b(v(t))\Vert '_{L^{\Phi ^*}(\Omega )} \le 2\sup _{t \in [0,T+h_o]} \Vert b(v(t))\Vert '_{L^{\Phi ^*}(\Omega )}. \end{aligned}$$

By Remark 2.4, this implies that the sequence \(\big (b(v(t+h))-b(v(t))\big )_{h \in (0,h_o)}\) is bounded in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\). Therefore, we find that

$$\begin{aligned} b(v(t+h))-b(v(t)) \mathop {\rightharpoondown }\limits ^{*}0 \text{ weakly }^* \text{ in } L^\infty (0,T;L^{\Phi ^*}(\Omega )) \text{ as } h \downarrow 0. \end{aligned}$$

Combining this with \(\Delta _h v \rightarrow \partial _t v\) in \(L^1(0,T;L^\Phi (\Omega ))\) as \(h \downarrow 0\), we infer

$$\begin{aligned} 0 \le \limsup _{h \downarrow 0}\varvec{\delta }_1(h)&= \limsup _{h \downarrow 0} \tfrac{1}{h} \iint _{\Omega _T} {\mathfrak {b}}\big [v(t),v(t+h)\big ] \,\mathrm {d}x\mathrm {d}t\nonumber \\&\le \lim _{h \downarrow 0} \iint _{\Omega _T} \big (b(v(t+h))-b(v(t))\big ) \Delta _h v(t) \,\mathrm {d}x\mathrm {d}t=0. \end{aligned}$$

Next, by the generalized Hölder’s inequality (1.8) and Hölder’s inequality, we compute that

$$\begin{aligned} |\varvec{\delta }_2(h)|&\le \left (\sup _{t \in [0,2h_o]} \Vert b(v(t))\Vert '_{L^{\Phi ^*}(\Omega )} +\sup _{t \in [0,h_o]} \Vert b(u(t))\Vert '_{L^{\Phi ^*}(\Omega )}\right ) \int _0^h \Vert \Delta _h v\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\\&\le \left (\sup _{t \in [0,2h_o]} \Vert b(v(t))\Vert '_{L^{\Phi ^*}(\Omega )} +\sup _{t \in [0,h_o]} \Vert b(u(t))\Vert '_{L^{\Phi ^*}(\Omega )}\right ) \int _0^h \Vert \partial _t v\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\rightarrow 0 \end{aligned}$$

as \(h \downarrow 0\). This concludes the proof of the lemma. \(\square\)

Mollification in time

In addition to standard mollification, we also consider the following mollification technique introduced by Landes [22]. We construct the regularization \([v]_h\), \(h>0\), to a given function v, such that it formally solves the ordinary differential equation

$$\begin{aligned} \partial _t [v]_h = -\tfrac{1}{h}\left( [v]_h-v\right) \end{aligned}$$
(2.6)

with initial condition \([v]_h(0) = v_o\). The precise construction is as follows. Let X be a separable Banach space and \(v_o \in X\); in the applications, we will have \(X=L^r(\Omega )\) with \(r\ge 1\) and \(X=L^\Phi (\Omega )\). Now, we consider \(v \in L^r(0,T;X)\) for some \(1 \le r \le \infty\), and define for \(h \in (0,T]\) and \(t \in [0,T]\) the mollification in time by

$$\begin{aligned}{}[v]_h(t) := e^{-\frac{t}{h}}v_o + \tfrac{1}{h} \int _0^t e^\frac{s-t}{h}v(s) \,\mathrm {d}s. \end{aligned}$$
(2.7)

It is easy to check that \([v]_h\) satisfies (2.6). The basic properties of the mollifications \([\cdot ]_h\) are provided in the following Lemma, cf. [10, Appendix B] for the proofs of the statements.

Lemma 2.15

Suppose that X is a separable Banach space and \(v_o \in X\). If \(v \in L^r(0,T;X)\) for some \(r \ge 1\), then the mollification \([v]_h\) defined in (2.7) fulfills \([v]_h \in L^r(0,T;X)\) and for any \(t_o \in (0,T]\) there holds

$$\begin{aligned} \left\| [v]_h\right\| _{L^r(0,t_o;X)} \le \left\| v\right\| _{L^r(0,t_o;X)} + \left[ \tfrac{h}{r}\left( 1-e^{-\frac{t_o r}{h}}\right) \right] ^\frac{1}{r} \left\| v_o\right\| _X. \end{aligned}$$

In the case \(r=\infty\) , the bracket \([\ldots ]^\frac{1}{r}\) in the preceding inequality has to be interpreted as 1. Moreover, in the case \(r<\infty\) we have \([v]_h \rightarrow v\) in \(L^r(0,T;X)\) as \(h \downarrow 0\). Finally, if \(v \in C^0([0,T];X)\) and \(v_o=v(0)\), then \([v]_h \in C^0([0,T];X)\), \([v]_h(0) =v_o\), and moreover \([v]_h \rightarrow v\) in \(C^0([0,T];X)\) as \(h \downarrow 0\). \(\square\)

For maps \(v \in L^r(0,T;X)\) with \(\partial _t v \in L^r(0,T;X)\) we have the following assertion.

Lemma 2.16

Let X be a separable Banach space and \(r \ge 1\). Assume that \(v \in L^r(0,T;X)\) with \(\partial _t v \in L^r(0,T;X)\). Then, for the mollification in time defined by

$$\begin{aligned}{}[v]_h(t) := e^{-\frac{t}{h}}v(0) + \tfrac{1}{h} \int _0^t e^\frac{s-t}{h}v(s) \,\mathrm {d}s\end{aligned}$$

, the time derivative can be computed by

$$\begin{aligned} \partial _t [v]_h(t) = \tfrac{1}{h} \int _0^t e^\frac{s-t}{h}\partial _t v(s) \,\mathrm {d}s, \end{aligned}$$

and, moreover we have that

$$\begin{aligned} \left\| \partial _t[v]_h\right\| _{L^r(0,T;X)} \le \left\| \partial _t v\right\| _{L^r(0,T;X)} \end{aligned}$$

holds true. \(\square\)

Lemma 2.17

Let \(r \ge 1\), \((v_i)_{i \in \mathbb {N}}\) be a sequence in \(L^r(\Omega _T)\), \(v \in L^r(\Omega _T)\) and \(v_o \in L^r(\Omega )\). If \(v_i \rightharpoondown v\) weakly in \(L^r(\Omega _T)\) as \(i \rightarrow \infty\), then \([v_i]_h \rightharpoondown [v]_h\) weakly in \(L^r(\Omega _T)\) as \(i \rightarrow \infty\) holds true for the mollifications defined by (2.7) with fixed \(h>0\) and initial values \(v_o\).

The initial condition

Here, we show that variational solutions attain the initial datum \(g_o\) in the \(L^\Phi\)-sense. For the proof of the following statement, we refer to [11, Lemma 2.9].

Lemma 2.18

Any variational solution to (1.2) in the sense of Definition 1.1fulfills the initial condition \(u(0)=g_o\) in the \(L^\Phi\)-sense, i.e.

$$\begin{aligned} \lim _{h \downarrow 0} \tfrac{1}{h} \varrho _{\Omega _h}(u - g_o) = 0. \end{aligned}$$

Proof

Since \(v=g\) is admissible in the variational inequality (1.11), by the generalized Hölder’s inequality (1.8), Lemmas 2.3 and 2.5, we find that

$$\begin{aligned}&\iint _{\Omega _\tau } f(Du) \,\mathrm {d}x\mathrm {d}t+ {\mathfrak {B}}[u(\tau ),g(\tau )] \\&\quad \le \sup _{t \in [0,T]} \big (\Vert b(g(t))\Vert '_{L^{\Phi ^*}(\Omega )} +\Vert b(u(t))\Vert '_{L^{\Phi ^*}(\Omega )} \big ) \int _0^\tau \Vert \partial _t g\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t+\iint _{\Omega _\tau } f(Dg) \,\mathrm {d}x\mathrm {d}t\\&\quad \le \sup _{t \in [0,T]} \big (2 + \varrho _\Omega ^*(b(g(t)))^\frac{1}{\ell +1} + \varrho _\Omega ^*(b(u(t)))^\frac{1}{\ell +1}\big ) \int _0^\tau \Vert \partial _t g\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\\&\qquad + \iint _{\Omega _\tau } f(Dg) \,\mathrm {d}x\mathrm {d}t\\&\quad \le \sup _{t \in [0,T]} \big (2 + (m \varrho _\Omega (g(t)))^\frac{1}{\ell +1} + (m \varrho _\Omega (u(t)))^\frac{1}{\ell +1} \big ) \int _0^\tau \Vert \partial _t g\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\\&\qquad + \iint _{\Omega _\tau } f(Dg) \,\mathrm {d}x\mathrm {d}t\end{aligned}$$

holds true for a.e. \(\tau \in [0,T]\). Recalling \(u,g \in L^\infty (0,T;L^\Phi (\Omega ))\) and (1.7), we have that

$$\begin{aligned} C := \sup _{t \in [0,T]} \big (2 + (m \varrho _\Omega (g(t)))^\frac{1}{\ell +1} + (m \varrho _\Omega (u(t)))^\frac{1}{\ell +1} \big ) < \infty . \end{aligned}$$

Furthermore, we know that \(\partial _t g \in L^1(0,T;L^\Phi (\Omega ))\). Altogether, discarding the nonnegative energy term on the left-hand side, taking the square root, integrating over \(\tau \in (0,h)\) for \(h \in (0,T)\) and dividing the result by h we conclude that

$$\begin{aligned} \tfrac{1}{h} \int _0^h {\mathfrak {B}}[u(\tau ),g(\tau )]^\frac{1}{2} \,\mathrm {d}\tau&\le \tfrac{1}{h} \int _0^h \left( \iint _{\Omega _\tau } f(Dg) \,\mathrm {d}x\mathrm {d}t+ C \int _0^\tau \Vert \partial _t g\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\right) ^\frac{1}{2} \,\mathrm {d}\tau \nonumber \\&\le \left( \iint _{\Omega _h} f(Dg) \,\mathrm {d}x\mathrm {d}t+ C \int _0^h \Vert \partial _t g\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\right) ^\frac{1}{2} \rightarrow 0 \end{aligned}$$
(2.8)

in the limit \(h \downarrow 0\). Next, from (2.3) and the convexity of \(\Phi\), we deduce that

$$\begin{aligned} \Phi \big (|u(t)-g_o|\big )&\le \Phi \left( 2 \left( \tfrac{1}{2}|u(t)-g(t)|+\tfrac{1}{2}|g(t)-g_o|\right) \right) \nonumber \\&\le 2^m \left[ \Phi \big (|u(t)-g(t)|\big ) +\Phi \big (|g(t)-g_o|\big )\right] \end{aligned}$$
(2.9)

holds true for a.e. \(t \in [0,T]\). Since \(g \in C^0([0,T];L^\Phi (\Omega ))\), we have that \(g(t) \rightarrow g_o\) strongly in \(L^\Phi (\Omega )\) as \(t \downarrow 0\) and hence by (1.6) that \(\Phi (|g(t)-g_o|) \rightarrow 0\) as \(t \downarrow 0\). Thus, we conclude that

$$\begin{aligned} \lim _{h \downarrow 0} \tfrac{1}{h} \varrho _{\Omega _h}(g - g_o) = 0. \end{aligned}$$
(2.10)

Hence, using the estimate from Lemma 2.9, Hölder’s inequality and Lemma 2.7, we infer

$$\begin{aligned} \tfrac{1}{h} \varrho _{\Omega _h}(u(t)-g(t))&\le \tfrac{1}{h} \iint _{\Omega _h} |\Phi (u(t))-\Phi (g(t))| \,\mathrm {d}x\mathrm {d}t\\&\le \tfrac{1}{h} \iint _{\Omega _h} \left| \sqrt{\Phi (u(t))} - \sqrt{\Phi (g(t))}\right| \left| \sqrt{\Phi (u(t))} + \sqrt{\Phi (g(t))}\right| \,\mathrm {d}x\mathrm {d}t\\&\le \tfrac{1}{h} \int _0^h \left\| \sqrt{\Phi (u(t))} - \sqrt{\Phi (g(t))}\right\| _{L^2(\Omega )} \\&\quad \left\| \sqrt{\Phi (u(t))} + \sqrt{\Phi (g(t))}\right\| _{L^2(\Omega )} \,\mathrm {d}t\\&\le \tfrac{1}{h} \sup _{t \in (0,h)} \big (\varrho _\Omega (u(t))^\frac{1}{2} + \varrho _\Omega (g(t))^\frac{1}{2}\big ) \\&\quad \int _0^h \left( \int _\Omega \left| \sqrt{\Phi (u(t))} - \sqrt{\Phi (g(t))}\right| ^2 \,\mathrm {d}x\right) ^\frac{1}{2} \,\mathrm {d}t\\&\le \tfrac{c(m,\ell )}{h} \sup _{t \in (0,h)} \big (\varrho _\Omega (u(t))^\frac{1}{2} + \varrho _\Omega (g(t))^\frac{1}{2}\big ) \int _0^h {\mathfrak {B}}[u(\tau ),g(\tau )]^\frac{1}{2} \,\mathrm {d}t. \end{aligned}$$

By \(u,g \in L^\infty (0,T;L^\Phi (\Omega ))\) and (1.7), the first term on the right-hand side of the preceding inequality is bounded. Therefore, combining the preceding inequality with (2.8), we find that

$$\begin{aligned} \lim _{h \downarrow 0} \tfrac{1}{h} \varrho _{\Omega _h}(u - g) = 0. \end{aligned}$$
(2.11)

Altogether, combining (2.9), (2.10) and (2.11), we obtain that

$$\begin{aligned} \tfrac{1}{h} \varrho _{\Omega _h}(u - g_o) \le \tfrac{2^m}{h} \varrho _{\Omega _h}(u - g) +\tfrac{2^m}{h} \varrho _{\Omega _h}(g - g_o) \rightarrow 0 \end{aligned}$$

in the limit \(h \downarrow 0\). This concludes the proof of the lemma. \(\square\)

Remark 2.19

In the case \(\Phi (v) = |v|^{m+1}\), i.e., \(L^\Phi (\Omega ) = L^{m+1}(\Omega )\), the statement of the preceding lemma reduces to the usual convergence assertion in the \(L^{m+1}\)-sense,

$$\begin{aligned} \lim _{h \downarrow 0} \tfrac{1}{h} \varrho _{\Omega _h}(u - g_o) = \lim _{h \downarrow 0} \tfrac{1}{h} \int _0^h \Vert u(t)-g_o\Vert _{L^{m+1}(\Omega )}^{m+1} \,\mathrm {d}t=0. \end{aligned}$$

An energy bound

In this section, we derive an energy bound for variational solutions.

Lemma 2.20

(Energy bound) Assume that (1.9) and (1.10) are satisfied and that u is a variational solution to (1.2) in the sense of Definition1.1. Then, u fulfills the energy bound

$$\begin{aligned}& \tfrac{1}{2} \sup _{t \in [0,T]} \varrho _\Omega (u(t)) + \iint _{\Omega _T} f(Du) \,\mathrm {d}x \le \tfrac{2^\ell }{m} \\&\quad + \tfrac{1}{2} \sup _{t \in [0,T]} \varrho _\Omega (v(t)) + \tfrac{3}{\ell } \iint _{\Omega _T} f(Dv) \,\mathrm {d}x\mathrm {d}t\\&\quad + c \bigg (\int _0^T \Vert \partial _t v\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\bigg )^\frac{\ell +1}{\ell } \\&\quad + \tfrac{3}{\ell } {\mathfrak {B}}[g_o,v(0)] \end{aligned}$$

with a constant \(c=c(\ell ,m)\) for any comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\).

Proof

If u is a variational solution and v an admissible comparison map in the sense of Definition 1.1, we have that

$$\begin{aligned}&\iint _{\Omega _\tau } f(Du) \,\mathrm {d}x\mathrm {d}t+{\mathfrak {B}}[u(\tau ),v(\tau )] \nonumber \\&\quad \le \iint _{\Omega _\tau } \partial _t v (b(v)-b(u)) \,\mathrm {d}x\mathrm {d}t+\iint _{\Omega _\tau } f(Dv) \,\mathrm {d}x\mathrm {d}t+{\mathfrak {B}}[g_o,v(0)] \end{aligned}$$
(2.12)

for a.e. \(\tau \in [0,T]\). By Lemmas 2.3 and 2.6, we obtain that

$$\begin{aligned} \varrho _\Omega (u(\tau )) \le \tfrac{1}{\ell } \varrho _\Omega ^*(b(u(\tau ))) \le \tfrac{2}{\ell } {\mathfrak {B}}[u(\tau ),v(\tau )] +\tfrac{2^{m+2}}{\ell } \varrho _\Omega (v(\tau )) \end{aligned}$$
(2.13)

holds true for any \(\tau \in [0,T]\). For \(\varepsilon >0\), by the generalized Hölder’s inequality (1.8), Hölder’s inequality, Lemma 2.5 applied to \(\Vert \cdot \Vert '_{L^{\Phi ^*}(\Omega )}\) and Young’s inequality, we infer

$$\begin{aligned} \iint _{\Omega _T}&|\partial _t v| |b(v)-b(u)| \,\mathrm {d}x\nonumber \\&\le \sup _{t \in [0,T]} \big (\Vert b(v(t))\Vert '_{L^{\Phi ^*}(\Omega )} +\Vert b(u(t))\Vert '_{L^{\Phi ^*}(\Omega )}\big ) \int _0^T \Vert \partial _t v\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\nonumber \\&\le \big (2 +\sup _{t \in [0,T]} \varrho _\Omega ^*(b(v(t)))^\frac{1}{\ell +1} +\sup _{t \in [0,T]} \varrho _\Omega ^*(b(u(t)))^\frac{1}{\ell +1}\big ) \int _0^T \Vert \partial _t v\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\nonumber \\&\le \big (2 +\sup _{t \in [0,T]} (m \varrho _\Omega (v(t)))^\frac{1}{\ell +1} +\sup _{t \in [0,T]} (m \varrho _\Omega (u(t)))^\frac{1}{\ell +1} \big ) \int _0^T \Vert \partial _t v\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\nonumber \\&\le 2^{\ell +1}\varepsilon + m\varepsilon \sup _{t \in [0,T]} \varrho _\Omega (v(t)) + m\varepsilon \sup _{t \in [0,T]} \varrho _\Omega (u(t)) \nonumber \\&\quad + c(\varepsilon ) \bigg (\int _0^T \Vert \partial _t v\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\bigg )^\frac{\ell +1}{\ell }. \end{aligned}$$
(2.14)

Inserting (2.13) and (2.14) into (2.12), we conclude that

$$\begin{aligned}&\sup _{\tau \in [0,T]} \varrho _\Omega (u(\tau )) + \iint _{\Omega _T} f(Du) \,\mathrm {d}x\mathrm {d}t\\&\quad \le \tfrac{2}{\ell } \sup _{\tau \in [0,T]} {\mathfrak {B}}[u(\tau ),v(\tau )] + \iint _{\Omega _T} f(Du) \,\mathrm {d}x\mathrm {d}t+\tfrac{2^{m+2}}{\ell } \sup _{\tau \in [0,T]} \varrho _\Omega (v(\tau ))\\&\quad \le \tfrac{3}{\ell }\left[ \iint _{\Omega _T} |\partial _t v| |b(v)-b(u)| \,\mathrm {d}x\mathrm {d}t+ \iint _{\Omega _T} f(Dv) \,\mathrm {d}x\mathrm {d}t+ {\mathfrak {B}}[g_o,v(0)]\right] \\&\qquad +\tfrac{2^{m+2}}{\ell } \sup _{\tau \in [0,T]} \varrho _\Omega (v(\tau ))\\&\quad \le \tfrac{3 \cdot 2^{\ell +1}}{\ell }\varepsilon + \tfrac{3m}{\ell } \varepsilon \sup _{t \in [0,T]} \varrho _\Omega (v(t)) + \tfrac{3m}{\ell } \varepsilon \sup _{t \in [0,T]} \varrho _\Omega (u(t))\\&\qquad + \tfrac{3}{\ell }c(\varepsilon ) \bigg (\int _0^T \Vert \partial _t v\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\bigg )^\frac{\ell +1}{\ell } + \tfrac{3}{\ell } \iint _{\Omega _T} f(Dv) \,\mathrm {d}x\mathrm {d}t+ \tfrac{3}{\ell } {\mathfrak {B}}[g_o,v(0)]. \end{aligned}$$

At this stage, we choose \(\varepsilon := \frac{\ell }{6m}\). This allows us to reabsorb the term

$$\begin{aligned} \tfrac{1}{2} \sup _{\tau \in [0,T]} \varrho _\Omega (u(\tau )) \end{aligned}$$

from the right-hand side of the preceding inequality into the left-hand side, which yields the claim. \(\square\)

Compactness

The proof of following result can be found in [11, Proposition 3.1].

Lemma 2.21

Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain, \(p>1\), \(T>0\) and \(k \in \mathbb {N}\). Suppose that for \(h_k := T/k\) piecewise constant maps \(u^{(k)} :\Omega \times (-h_k,T] \rightarrow \mathbb {R}_{\ge 0}\) are defined by

$$\begin{aligned} u^{(k)}(\cdot ,t) := u_i^{(k)} \text{ for } t \in ((i-1)h_k,ih_k] \text{ with } i \in \{0,\ldots ,k\} \end{aligned}$$

with nonnegative functions \(u_i^{(k)} \in L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\). Suppose further that there exists a constant \(M>0\) such that the energy estimate

$$\begin{aligned} \max _{i \in \{0,\ldots ,k\}} \left[ \varrho _\Omega \big (u_i^{(k)}\big ) + \int _\Omega \big |Du_i^{(k)}\big |^p \,\mathrm {d}x\right] \le M \end{aligned}$$

and the continuity estimate

$$\begin{aligned} \tfrac{1}{h_k} \sum _{i=1}^k \int _\Omega \left| \sqrt{\Phi \big (u_i^{(k)}\big )} -\sqrt{\Phi \big (u_{i-1}^{(k)}\big )} \right| ^2 \,\mathrm {d}x\le M \end{aligned}$$

hold true for all \(k \in \mathbb {N}\), and that \(u^{(k)} \rightharpoondown u\) weakly in \(L^p(0,T;W^{1,p}(\Omega ))\) as \(k \rightarrow \infty\). Then, there exists a subsequence \({\mathfrak {K}} \subset \mathbb {N}\) such that in the limit \({\mathfrak {K}} \ni k \rightarrow \infty\) we have the following convergences:

$$\begin{aligned} \left\{ \begin{array}{ll} \sqrt{\Phi \big (u^{(k)}\big )} \rightarrow \sqrt{\Phi (u)} &{}\text {strongly in } L^1(\Omega _T), \\ u^{(k)} \rightarrow u &{}\text {a.e.~in } \Omega _T. \end{array} \right. \end{aligned}$$

Existence for regular data

First, we prove an existence result for the case of regular boundary values and obstacle. More precisely, we consider nonnegative regular boundary values given by

$$\begin{aligned} \left\{ \begin{array}{l} g \in L^p(0,T;W^{1,p}(\Omega )), \; \partial _t g \in L^2(\Omega _T) \cap L^p(0,T;W^{1,p}(\Omega )) ,\\ g_o = g(0) \in L^2(\Omega ) \cap W^{1,p}(\Omega ), \kappa \le g \end{array} \right. \end{aligned}$$
(3.1)

with a constant \(\kappa >0\) and a nonnegative obstacle function \(\psi\) satisfying

$$\begin{aligned} \left\{ \begin{array}{l} \psi \in g + L^p(0,T;W^{1,p}_0(\Omega )), \; \partial _t \psi \in L^2(\Omega _T) \cap L^p(0,T;W^{1,p}(\Omega )), \\ \psi (0) \in L^2(\Omega ) \cap \big (g_o+W^{1,p}_0(\Omega )\big ), \; \kappa \le \psi \le g \; \text {a.e. on } \Omega _T. \end{array} \right. \end{aligned}$$
(3.2)

Theorem 3.1

Assume that the obstacle satisfies (3.2) and boundary values are given by (3.1). Then, there exists a variational solution to (1.2) in the sense of Definition 1.1.

A sequence of minimizers to elliptic variational functionals

Fix a step size \(h \in (0,1]\) such that \(h = T/K\) for some \(K \in \mathbb {N}\). For \(i \in \{0,\ldots ,K\}\) define \(g_i := g(ih)\) and \(\psi _i := \psi (ih)\). Set \(u_0 = g_o\). Then, we inductively find minimizers \(u_i\) of the functionals

$$\begin{aligned} F_i[v] = \int _\Omega f(Dv) \,\mathrm {d}x+ \tfrac{1}{h}\int _\Omega {\mathfrak {b}}[u_{i-1},v] \,\mathrm {d}x\end{aligned}$$

in the class \(v \in L^\Phi (\Omega ) \cap \big (g_i+W^{1,p}_0(\Omega )\big )\) with \(v \ge \psi _i\). Note that this class is not empty, since \(v=g_i\) is admissible. The existence of minimizers \(u_i\) is ensured, for example, by the direct method of the calculus of variations. For convenience of the reader, we give the precise proof.

Proposition 3.2

Assume that nonnegative functions \(g_*\in L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\), \(\psi _*\in L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\) and \(u_*\in L^\Phi (\Omega )\) are given. Then, there exists a minimizer u of

$$\begin{aligned} F[v] = \int _\Omega f(Dv) \,\mathrm {d}x+ \tfrac{1}{h}\int _\Omega {\mathfrak {b}}[u_*,v] \,\mathrm {d}x\end{aligned}$$

in the class of functions \(v \in L^\Phi (\Omega ) \cap \big (g_*+W^{1,p}_0(\Omega )\big )\) with \(v \ge \psi _*\).

Proof

Consider a minimizing sequence \((u_j)_{j \in \mathbb {N}}\), i.e.

$$\begin{aligned} \lim _{j \rightarrow \infty } F[u_j] = \inf \big \{F[v] :v \in L^\Phi (\Omega ) \cap \big (g_*+W^{1,p}_0(\Omega )\big ) \;\text {with } v \ge \psi _*\big \}. \end{aligned}$$

By means of Lemmas 2.3 and 2.6, we find that

$$\begin{aligned} \Phi (u_j) \le 2 {\mathfrak {b}}[u_*,u_j] + 2^{2+\frac{1}{\ell }} \Phi ^*(b(u_*)) \le 2 {\mathfrak {b}}[u_*,u_j] + 2^{2+\frac{1}{\ell }} m \Phi (u_*). \end{aligned}$$

Together with (1.3) and the fact that \(h \in (0,1]\), this implies

$$\begin{aligned} \int _\Omega \nu |Du_j|^p \,\mathrm {d}x+ \tfrac{1}{2} \varrho _\Omega (u_j)&\le \int _\Omega \big [f(Du_j) + {\mathfrak {b}}[u_*,u_j] \big ] \,\mathrm {d}x+ 2^{1+\frac{1}{\ell }} m \varrho _\Omega (u_*) \\&\le F[u_j] + 2^{1+\frac{1}{\ell }} m \varrho _\Omega (u_*). \end{aligned}$$

Hence, by (1.7) the sequence \((u_j)_{j \in \mathbb {N}}\) in bounded in \(L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\). Thus, there exists a (not relabeled) subsequence and a limit map \(u \in L^\Phi (\Omega ) \cap \big (g_*+W^{1,p}_0(\Omega )\big )\) such that

$$\begin{aligned} \left\{ \begin{array}{l} u_j \rightharpoondown u \text{ weakly in } L^\Phi (\Omega ), \\ u_j \rightharpoondown u \hbox { weakly in } W^{1,p}(\Omega ) \end{array} \right. \end{aligned}$$

in the limit \(j \rightarrow \infty\). Observe that the obstacle condition \(u \ge \psi _*\) a.e. in \(\Omega\) is preserved. Since F is convex and lower semicontinuous with respect to strong convergence in \(L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\) by means of Fatou’s Lemma, F is also lower semicontinuous with respect to weak convergence in \(L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\), cf. [14, Corollary 3.9]. As a consequence, we find that

$$\begin{aligned} F[u] \le \lim _{j \rightarrow \infty } F[u_j] = \inf \big \{F[v] :v \in L^\Phi (\Omega ) \cap \big (g_*+W^{1,p}_0(\Omega )\big ) \;\text {with } v \ge \psi _*\big \}, \end{aligned}$$

which yields the claim. \(\square\)

Energy estimates

Observe that \(v := u_{i-1} + \psi _i - \psi _{i-1} \ge \psi _i\) is an admissible comparison function for \(u_i\). Using the minimality of \(u_i\) with respect to \(F_i\), we obtain that

$$\begin{aligned} \int _\Omega&f(Du_i) \,\mathrm {d}x+ \tfrac{1}{h} \int _\Omega {\mathfrak {b}}[u_{i-1},u_i] \,\mathrm {d}x= F_i[u_i] \\&\le F_i[u_{i-1} + \psi _i - \psi _{i-1}] \\&= \int _\Omega f(D(u_{i-1} + \psi _i - \psi _{i-1})) \,\mathrm {d}x+ \tfrac{1}{h} \int _\Omega {\mathfrak {b}}[u_{i-1},u_{i-1} + \psi _i - \psi _{i-1}] \,\mathrm {d}x\\&=: \mathrm {I} + \mathrm {II}, \end{aligned}$$

where the definition of \(\mathrm {I}\) and \(\mathrm {II}\) is clear in this context. First, we estimate \(\mathrm {I}\). To this end, using the Lipschitz estimate (1.4), Young’s inequality, the assumption \(h \le 1\) and the coercivity assumption (1.3), we conclude that

$$\begin{aligned} |f(&D(u_{i-1} + \psi _i - \psi _{i-1})) - f(Du_{i-1})| \\&\le c \big (1 + |D(u_{i-1} + \psi _i - \psi _{i-1})|^{p-1} + |Du_{i-1}|^{p-1}\big ) |D\psi _i - D\psi _{i-1}| \\&\le c |Du_{i-1}|^{p-1} |D\psi _i - D\psi _{i-1}| + c\big (1+|D\psi _i - D\psi _{i-1}|^{p-1}\big ) |D\psi _i - D\psi _{i-1}| \\&\le \nu h |Du_{i-1}|^p + c\big (1+h^{1-p}|D\psi _i - D\psi _{i-1}|^{p-1}\big ) |D\psi _i - D\psi _{i-1}| \\&\le h f(Du_{i-1}) + c\big (1+h^{1-p}|D\psi _i - D\psi _{i-1}|^{p-1}\big ) |D\psi _i - D\psi _{i-1}| \end{aligned}$$

holds true with a constant \(c=c(p,n,L,\nu )\). Moreover, we find that

$$\begin{aligned} \int _\Omega |D\psi _i - D\psi _{i-1}| \,\mathrm {d}x\le \iint _{\Omega \times ((i-1)h,ih)} |\partial _t D\psi | \,\mathrm {d}x\mathrm {d}t\end{aligned}$$

and that

$$\begin{aligned} \int _\Omega |D\psi _i - D\psi _{i-1}|^p \,\mathrm {d}x\le h^{p-1} \iint _{\Omega \times ((i-1)h,ih)} |\partial _t D\psi |^p \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

Therefore, we deduce the estimate

$$\begin{aligned} \mathrm {I} \le (1+h) \int _\Omega f(Du_{i-1}) \,\mathrm {d}x+ c \iint _{\Omega \times ((i-1)h,ih)} \big [|\partial _t D\psi | + |\partial _t D\psi |^p\big ] \,\mathrm {d}x\mathrm {d}t\end{aligned}$$

with a constant \(c=c(n,p,L,\nu )\). Next, by Lemma 2.7 and (1.5) together with Remark 2.2 and the fact that \(u_{i-1} \ge \psi _{i-1} \ge \kappa\) and \(0<\ell \le m \le 1\), we estimate \(\mathrm {II}\), which leads to

$$\begin{aligned} {\mathfrak {b}}[u_{i-1},u_{i-1}+\psi _i-\psi _{i-1}]&= (b(u_{i-1}+\psi _i-\psi _{i-1}) - b(u_{i-1})) (\psi _i - \psi _{i-1}) \\&= \int _{u_{i-1}+\psi _i-\psi _{i-1}}^{u_{i-1}} b'(s) \,\mathrm {d}s\cdot (\psi _i - \psi _{i-1}) \\&\le m b(1) \max \{\kappa ^{\ell -1},\kappa ^{m-1}\} (\psi _i - \psi _{i-1})^2. \end{aligned}$$

Therefore, by Hölder’s inequality, we conclude that

$$\begin{aligned} \mathrm {II}&\le \tfrac{m b(1)}{h} \max \{\kappa ^{\ell -1},\kappa ^{m-1}\} \int _\Omega (\psi _i-\psi _{i-1})^2 \,\mathrm {d}x\\&= \tfrac{m b(1)}{h} \max \{\kappa ^{\ell -1},\kappa ^{m-1}\} \int _\Omega \bigg |\int _{(i-1)h}^{ih} \partial _t \psi \,\mathrm {d}t\bigg |^2 \,\mathrm {d}x\\&\le m b(1) \max \{\kappa ^{\ell -1},\kappa ^{m-1}\} \iint _{\Omega \times ((i-1)h,ih)} |\partial _t \psi |^2 \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

Combining the estimates for \(\mathrm {I}\) and \(\mathrm {II}\), we obtain that

$$\begin{aligned} \int _\Omega f(Du_i) \,\mathrm {d}x&+ \tfrac{1}{h} \int _\Omega {\mathfrak {b}}[u_{i-1},u_i] \,\mathrm {d}x\nonumber \\&\le (1+h) \int _\Omega f(Du_{i-1}) \,\mathrm {d}x\nonumber \\&\quad + c \iint _{\Omega \times ((i-1)h,ih)} \big [|\partial _t \psi |^2 +|\partial _t D\psi | + |\partial _t D\psi |^p\big ] \,\mathrm {d}x\mathrm {d}t\end{aligned}$$
(3.3)

holds true with a constant \(c=c(\kappa ,b(1),\ell ,m,p,n,L,\nu )\). Summing up (3.3) from \(i=1,\ldots ,k\) for some \(k \in \{1,\ldots ,K\}\) leads to

$$\begin{aligned} \sum _{i=1}^k&\int _\Omega f(Du_i) \,\mathrm {d}x+ \tfrac{1}{h} \sum _{i=1}^k \int _\Omega {\mathfrak {b}}[u_{i-1},u_i] \,\mathrm {d}x\\&\le (1+h) \sum _{i=1}^k \int _\Omega f(Du_{i-1}) \,\mathrm {d}x+ c \iint _{\Omega \times (0,kh)} \big [|\partial _t \psi |^2 +|\partial _t D\psi | + |\partial _t D\psi |^p\big ] \,\mathrm {d}x\mathrm {d}t\\&\le (1+h) \sum _{i=1}^{k-1} \int _\Omega f(Du_i) \,\mathrm {d}x+ (1+h) \varvec{\Psi }(kh), \end{aligned}$$

where we used the short-hand notation

$$\begin{aligned} \varvec{\Psi }(\tau ) = L\int _\Omega \big [1+|Dg_o|^p\big ] \,\mathrm {d}x+c \iint _{\Omega \times (0,\tau )} \big [|\partial _t \psi |^2 +|\partial _t D\psi | + |\partial _t D\psi |^p\big ] \,\mathrm {d}x\mathrm {d}t\end{aligned}$$

for any \(\tau \in (0,T]\). Reabsorbing \(\sum _{i=1}^{k-1} \int _\Omega f(Du_i) \,\mathrm {d}x\) into the left-hand side of the preceding inequality, we have that

$$\begin{aligned} \int _\Omega f(Du_k) \,\mathrm {d}x&+ \tfrac{1}{h} \sum _{i=1}^k \int _\Omega {\mathfrak {b}}[u_{i-1},u_i] \,\mathrm {d}x\nonumber \\&\le h \sum _{i=1}^{k-1} \int _\Omega f(Du_i) \,\mathrm {d}x+(1+h) \varvec{\Psi }(kh). \end{aligned}$$
(3.4)

In order to estimate the right-hand side of the preceding inequality, we iterate (3.3) from \(j=1,\ldots ,i\) for \(i=1,\ldots ,k-1\). This yields

$$\begin{aligned} \int _\Omega f(Du_i) \,\mathrm {d}x&\le (1+h)^i \int _\Omega f(Dg_o) \,\mathrm {d}x\\&\quad +\sum _{j=1}^i c\, (1+h)^{i-j} \iint _{\Omega \times ((j-1)h,jh)} \big [|\partial _t \psi |^2 +|\partial _t D\psi | + |\partial _t D\psi |^p\big ] \,\mathrm {d}x\mathrm {d}t\\&\le (1+h)^i \varvec{\Psi }(ih) \le (1+h)^i \varvec{\Psi }(kh). \end{aligned}$$

Inserting this into (3.4), we obtain that

$$\begin{aligned} \int _\Omega f(Du_k) \,\mathrm {d}x+ \tfrac{1}{h} \sum _{i=1}^k \int _\Omega {\mathfrak {b}}[u_{i-1},u_i] \,\mathrm {d}x&\le h \sum _{i=1}^{k-1} (1+h)^i \varvec{\Psi }(kh) +(1+h) \varvec{\Psi }(kh) \\&= (1+h)^k \varvec{\Psi }(kh). \end{aligned}$$

Since \((1+h)^k \le (1+h)^K = (1+\frac{T}{K})^K \le e^T\), we conclude that

$$\begin{aligned} \int _\Omega f(Du_k) \,\mathrm {d}x+ \tfrac{1}{h} \sum _{i=1}^k \int _\Omega {\mathfrak {b}}[u_{i-1},u_i] \,\mathrm {d}x\le e^T \varvec{\Psi }(T) \end{aligned}$$
(3.5)

for any \(k \in \mathbb {N}\) with \(kh \le T\). Finally, from Lemma 2.7 and (3.5), we infer the estimate

$$\begin{aligned} \varrho _\Omega (u_k)&\le 2k \sum _{i=1}^k \int _\Omega \big |\sqrt{\Phi (u_i)} - \sqrt{\Phi (u_{i-1})}\big |^2 \,\mathrm {d}x+ 2 \varrho _\Omega (g_o) \nonumber \\&\le 2 k c^2(\ell ,m) \sum _{i=1}^k \int _\Omega {\mathfrak {b}}[u_{i-1},u_i] \,\mathrm {d}x+ 2 \varrho _\Omega (g_o) \nonumber \\&\le 2 k h c e^T \varvec{\Psi }(T) + 2 \varrho _\Omega (g_o) \nonumber \\&\le 2 T c e^T \varvec{\Psi }(T) + 2 \varrho _\Omega (g_o) \end{aligned}$$
(3.6)

for some constant \(c=c(\kappa ,b(1),\ell ,m,p,n,L,\nu )\).

Convergence to a limit map

In the following, we set \(h_K := T/K\) for \(K \in \mathbb {N}\). We define the piecewise constant function \(u^{(K)} :\Omega \times (-h_K,T] \rightarrow \mathbb {R}_{\ge 0}\) by

$$\begin{aligned} u^{(K)}(\cdot ,t) := u_i \text{ for } t \in ((i-1)h_K,ih_K] \text{ with } i \in \{0,\ldots ,K\}. \end{aligned}$$

Analogously, we define \(g^{(K)}\) and \(\psi ^{(K)}\). Combining estimate (3.5) with the coercivity condition (1.3)\(_1\) and discarding the nonnegative sum on the left-hand side, we find that

$$\begin{aligned} \nu \sup _{t \in [0,T]} \int _\Omega \big |Du^{(K)}(t)\big |^p \,\mathrm {d}x\le e^T \varvec{\Psi }(T). \end{aligned}$$

Hence, the sequence \((u^{(K)})_{K \in \mathbb {N}}\) is bounded in \(L^\infty (0,T;W^{1,p}(\Omega ))\). Since \(u^{(K)} \in g^{(K)} + L^\infty (0,T;W^{1,p}_0(\Omega ))\), there exists a subsequence \({\mathfrak {K}} \subset \mathbb {N}\) and a limit map \(u \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) such that

$$\begin{aligned} \hbox {} u^{(K)} \mathop {\rightharpoondown }\limits ^{*}u \hbox { weakly} ^* \text{ in } L^\infty (0,T;W^{1,p}(\Omega )) \end{aligned}$$
(3.7)

in the limit \({\mathfrak {K}} \ni K \rightarrow \infty\). By Lemma 2.7 and the energy estimates (3.5) and (3.6), the assumptions of Lemma 2.21 are satisfied for the sequence \((u^{(K)})_{K \in {\mathfrak {K}}}\). Therefore, choosing another subsequence still denoted by \({\mathfrak {K}}\), we obtain that

$$\begin{aligned} \left\{ \begin{array}{ll} \sqrt{\Phi \big (u^{(K)}\big )} \rightarrow \sqrt{\Phi (u)} &{}\text {strongly in } L^1(\Omega _T), \\ u^{(K)} \rightarrow u &{}\text {a.e.~in } \Omega _T \end{array} \right. \end{aligned}$$
(3.8)

in the limit \({\mathfrak {K}} \ni K \rightarrow \infty\). At this stage, observe that \(\psi ^{(K)} \rightarrow \psi\) a.e. in \(\Omega _T\) as \(k \rightarrow \infty\). Combining this with (3.8)\(_2\) and the fact that \(u^{(K)} \ge \psi ^{(K)}\) a.e. in \(\Omega _T\), we deduce that u satisfies the obstacle condition \(u \ge \psi\) a.e. in \(\Omega _T\). Next, by means of Lemma 2.7 and (3.5), we conclude that

$$\begin{aligned} \iint _{\Omega _T} \big |\Delta _{h_K} \sqrt{\Phi \big (u^{(K)}\big )}\big |^2 \,\mathrm {d}x\mathrm {d}t&= \tfrac{1}{h_K} \sum _{i=1}^K \int _\Omega \big |\sqrt{\Phi (u_i)}-\sqrt{\Phi (u_{i-1})}\big |^2 \,\mathrm {d}x\\&\le \tfrac{c^2(\ell ,m)}{h_K} \sum _{i=1}^K \int _\Omega {\mathfrak {b}}[u_{i-1},u_i] \,\mathrm {d}x\le c^2(\ell ,m) e^T \varvec{\Psi }(T). \end{aligned}$$

Thus, we find a subsequence such that \(\Delta _{h_K} \sqrt{\Phi \big (u^{(K)}\big )} \rightharpoondown w\) weakly in \(L^2(\Omega _T)\). In order to characterize w, we use this fact together with (3.8)\(_1\). More precisely, we obtain for any \(\varphi \in C^\infty _0(\Omega _T)\) that

$$\begin{aligned} \iint _{\Omega _T} \sqrt{\Phi (u)}\, \partial _t \varphi \,\mathrm {d}x\mathrm {d}t&= \lim _{{\mathfrak {K}} \ni K \rightarrow \infty } \iint _{\Omega _T} \sqrt{\Phi \big (u^{(K)}\big )}\, \Delta _{-h_K} \varphi \,\mathrm {d}x\mathrm {d}t\\&= -\lim _{{\mathfrak {K}} \ni K \rightarrow \infty } \iint _{\Omega _T} \Delta _{h_K} \sqrt{\Phi \big (u^{(K)}\big )}\, \varphi \,\mathrm {d}x\mathrm {d}t\\&= - \iint _{\Omega _T} w \varphi \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

By a density argument, this ensures that \(w = \partial _t \sqrt{\Phi (u)}\) and in particular \(\partial _t \sqrt{\Phi (u)} \in L^2(\Omega _T)\). Hence, we have that \(\partial _t \Phi (u) = 2\sqrt{\Phi (u)}\partial _t \sqrt{\Phi (u)} \in L^1(\Omega _T)\). This implies that \(\Phi (u) \in C^0([0,T];L^1(\Omega ))\) and hence by means of Lemma 2.9 that \(u \in C^0([0,T];L^\Phi (\Omega ))\).

Minimizing property of the approximations

Observe that for each \(K \in \mathbb {N}\), the map \(u^{(K)}\) is a minimizer of the functional

$$\begin{aligned} {\mathbf {F}}^{(K)}[v] := \iint _{\Omega _T} f(Dv) \,\mathrm {d}x\mathrm {d}t+ \tfrac{1}{h_K} \iint _{\Omega _T} {\mathfrak {b}}\big [u^{(K)}(t-h_K),v(t)\big ] \,\mathrm {d}x\mathrm {d}t\end{aligned}$$

in the class of functions \(v \in L^\Phi (\Omega _T) \cap \big (g^{(K)} + L^p(0,T;W^{1,p}_0(\Omega ))\big )\) satisfying \(v \ge \psi ^{(K)}\) a.e. in \(\Omega _T\). Indeed, using the definitions of \({\mathbf {F}}^{(K)}\) and \(u^{(K)}\) and the minimality of \(u_i\) with respect to \(F_i\), we compute for any admissible function v as above

$$\begin{aligned} {\mathbf {F}}^{(K)}\big [u^{(K)}\big ]&= \iint _{\Omega _T} f\big (Du^{(K)}\big ) \,\mathrm {d}x\mathrm {d}t+ \tfrac{1}{h_K} \iint _{\Omega _T} {\mathfrak {b}}\big [u^{(K)}(t-h_K),u^{(K)}(t)\big ] \,\mathrm {d}x\mathrm {d}t\\&= \sum _{i=1}^K \iint _{\Omega \times ((i-1)h_K,ih_K)} \left[ f(Du_i) +\tfrac{1}{h_K} {\mathfrak {b}}[u_{i-1},u_i]\right] \,\mathrm {d}x\mathrm {d}t\\&= \sum _{i=1}^K \int _{(i-1)h_K}^{ih_K} F_i[u_i] \,\mathrm {d}t\le \sum _{i=1}^K \int _{(i-1)h_K}^{ih_K} F_i[v] \,\mathrm {d}t\\&= \iint _{\Omega _T} f(Dv) \,\mathrm {d}x\mathrm {d}t+ \tfrac{1}{h_K} \iint _{\Omega _T} {\mathfrak {b}}\big [u^{(K)}(t-h_K),v(t)\big ] \,\mathrm {d}x\mathrm {d}t\\&= {\mathbf {F}}^{(K)}[v]. \end{aligned}$$

Note that for any fixed comparison map \(v \in L^\Phi (\Omega _T) \cap \big (g^{(K)} + L^p(0,T;W^{1,p}_0(\Omega ))\big )\) with \(v \ge \psi ^{(K)}\) a.e. in \(\Omega _T\) and any \(s \in (0,1)\) the convex combination \(w_s := u^{(K)} + s(v-u^{(K)})\) of \(u^{(K)}\) and v is still admissible, since \(\psi ^{(K)} \le w_s \in L^\Phi (\Omega _T) \cap \big (g^{(K)} + L^p(0,T;W^{1,p}_0(\Omega ))\big )\). Then, the minimality of \(u^{(K)}\) and the convexity of f imply that

$$\begin{aligned} {\mathbf {F}}^{(K)}\big [u^{(K)}\big ]&\le {\mathbf {F}}^{(K)}[w_s] \\&\le \iint _{\Omega _T} \left[ (1-s) f\big (Du^{(K)}\big ) + sf(Dv) +\tfrac{1}{h_K} {\mathfrak {b}}\big [u^{(K)}(t-h_K),w_s\big ]\right] \,\mathrm {d}x\mathrm {d}t\end{aligned}$$

for any \(s \in (0,1)\), with equality for \(s=0\). Reabsorbing \(\iint _{\Omega _T} (1-s) f\big (Du^{(K)}\big ) \,\mathrm {d}x\mathrm {d}t\) into the left-hand side of the preceding inequality and dividing by s, we find that

$$\begin{aligned}&\iint _{\Omega _T} f\big (Du^{(K)}\big ) \,\mathrm {d}x\mathrm {d}t\\&\quad \le \iint _{\Omega _T} \left[ f(Dv) +\tfrac{1}{s h_K} \big [{\mathfrak {b}}\big [u^{(K)}(t-h_K),w_s\big ] -{\mathfrak {b}}\big [u^{(K)}(t-h_K),u^{(K)}(t)\big ]\big ]\right] \,\mathrm {d}x\mathrm {d}t\\&\quad = \iint _{\Omega _T} \left[ f(Dv) +\tfrac{1}{h_K} \big [\tfrac{1}{s}\big (\Phi (w_s) - \Phi \big (u^{(K)}\big )\big ) -b\big (u^{(K)}(t-h_K\big ) \big (v-u^{(K)}\big )\big ]\right] \,\mathrm {d}x\mathrm {d}t\end{aligned}$$

holds true. Note that the map \(s \mapsto \tfrac{1}{s}\big (\Phi (w_s) - \Phi \big (u^{(K)}\big )\big )\) is monotone and converges a.e. in \(\Omega _T\) to the \(L^1(\Omega _T)\)-function \(b\big (u^{(K)}\big ) \big (v-u^{(K)}\big )\), since \(\Phi\) is convex. Passing to the limit \(s \downarrow 0\) in the preceding inequality with the aid of the dominated convergence theorem, we deduce that

$$\begin{aligned} \iint _{\Omega _T}&f\big (Du^{(K)}\big ) \,\mathrm {d}x\mathrm {d}t\\&\le \iint _{\Omega _T} \left[ f(Dv) +\tfrac{1}{h_K} \big [b\big (u^{(K)}\big ) - b\big (u^{(K)}(t-h_K)\big )\big ] \big (v-u^{(K)}\big )\right] \,\mathrm {d}x\mathrm {d}t\\&= \iint _{\Omega _T} \left[ f(Dv) + \Delta _{-h_K} b\big (u^{(K)}\big ) \big (v-u^{(K)}\big )\right] \,\mathrm {d}x\mathrm {d}t\end{aligned}$$

for any map \(v \in L^\Phi (\Omega _T) \cap \big (g^{(K)} + L^p(0,T;W^{1,p}_0(\Omega ))\big )\) with \(v \ge \psi ^{(K)}\) a.e. in \(\Omega _T\). Note that in particular \(u^{(K)}(t) = g_o\) for \(t \in (-h_K,0]\). Using the comparison map \(\chi _{(0,T)} v + \chi _{[\tau ,T]} u^{(K)}\) for some \(\tau \in (0,T]\) instead of v, we infer the localized variational inequality

$$\begin{aligned} \iint _{\Omega _\tau }&f\big (Du^{(K)}\big ) \,\mathrm {d}x\mathrm {d}t\le \iint _{\Omega _\tau } \left[ f(Dv) + \Delta _{-h_K} b\big (u^{(K)}\big ) \big (v-u^{(K)}\big )\right] \,\mathrm {d}x\mathrm {d}t \end{aligned}$$
(3.9)

for any \(\tau \in (0,T]\) and \(v \in L^\Phi (\Omega _\tau ) \cap \big (g^{(K)} + L^p(0,\tau ;W^{1,p}_0(\Omega ))\big )\) with \(v \ge \psi ^{(K)}\) a.e. in \(\Omega _\tau\).

Variational inequality for the limit map

Here, we pass to the limit \(K \rightarrow \infty\) in (3.9) in order to deduce the variational inequality for the limit map. To this end, we consider an arbitrary map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). We extend v to negative times by setting \(v(t) = v(0) \in L^\Phi (\Omega )\) for \(t<0\). Observe that the function \(v_K := v + \psi ^{(K)} - \psi\) is an admissible comparison map in (3.9), since \(v_k \ge \psi ^{(K)}\) and \(v_K \in L^\Phi (\Omega _T) \cap \big (g^{(K)} + L^p(0,T;W^{1,p}_0(\Omega ))\big )\) are satisfied. Hence, we obtain that

$$\begin{aligned} \iint _{\Omega _\tau } f\big (Du^{(K)}\big ) \,\mathrm {d}x\mathrm {d}t&\le \iint _{\Omega _\tau } \left[ f(Dv_K) + \Delta _{-h_K} b\big (u^{(K)}\big ) \big (v-u^{(K)}\big ) \right] \,\mathrm {d}x\mathrm {d}t\nonumber \\&\quad + \iint _{\Omega _\tau } \Delta _{-h_K} b\big (u^{(K)}\big )\big (\psi ^{(K)}-\psi \big ) \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$
(3.10)

Now, we consider the terms of (3.10) separately. First, by (3.7) and since f is convex and satisfies the coercivity condition (1.3)\(_1\), we have that

$$\begin{aligned} \iint _{\Omega _\tau } f(Du) \,\mathrm {d}x\mathrm {d}t\le \liminf _{{\mathfrak {K}} \ni K \rightarrow \infty } \iint _{\Omega _\tau } f\big (Du^{(K)}\big ) \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$
(3.11)

By (1.4) and the fact that \(D\psi ^{(K)} \rightarrow D\psi\) in \(L^p(\Omega _T,\mathbb {R}^n)\), we conclude that

$$\begin{aligned} \bigg |\iint _{\Omega _\tau }&f(Dv_K) - f(Dv) \,\mathrm {d}x\mathrm {d}t\bigg | \nonumber \\&\le c(n,p,L) \iint _{\Omega _\tau } \big (1 + \big |D\psi ^{(K)}-D\psi \big |^{p-1} + |Dv|^{p-1}\big ) \big |D\psi ^{(K)} - D\psi \big | \,\mathrm {d}x\mathrm {d}t\nonumber \\&\rightarrow 0 \end{aligned}$$
(3.12)

in the limit \(K \rightarrow \infty\). Next, shifting the difference quotient in the last term on the right-hand side from \(b\big (u^{(K)}\big )\) to \(\psi ^{(K)}-\psi\), we obtain that

$$\begin{aligned} \iint _{\Omega _\tau }&\Delta _{-h_K} b\big (u^{(K)}\big ) \big (\psi ^{(K)}-\psi \big ) \,\mathrm {d}x\mathrm {d}t\nonumber \\&= \iint _{\Omega \times (0,\tau -h_K]} b\big (u^{(K)}\big ) \Delta _{h_K}\big (\psi ^{(K)}-\psi \big ) \,\mathrm {d}x\mathrm {d}t\nonumber \\&\quad - \tfrac{1}{h_K} \iint _{\Omega \times (\tau -h_K,\tau )} b\big (u^{(K)}\big ) \big (\psi ^{(K)}-\psi \big ) \,\mathrm {d}x\mathrm {d}t\nonumber \\&\quad + \tfrac{1}{h_K} \iint _{\Omega \times (-h_K,0]} b\big (u^{(K)}\big ) \big (\psi ^{(K)}(t+h_K)-\psi (t+h_K)\big ) \,\mathrm {d}x\mathrm {d}t\nonumber \\&=: \mathrm {I}_K + \mathrm {II}_K + \mathrm {III}_K, \end{aligned}$$
(3.13)

where the definition of \(\mathrm {I}_K - \mathrm {III}_K\) is clear in this context. By the generalized Hölder’s inequality (1.8) and Hölder’s inequality, we find that

$$\begin{aligned} |\mathrm {I}_K| \le \sup _{t \in [0,\tau ]} \Vert b\big (u^{(K)}(t)\big )\Vert '_{L^{\Phi ^*}(\Omega )} \int _0^{\tau -h_K} \Vert \Delta _{h_K} v^{(K)} - \Delta _{h_K} v\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t. \end{aligned}$$

Combining the energy estimate (3.6) with Lemma 2.3 and recalling (1.7), we conclude that \(\big (b\big (u^{(K)}(t)\big )_{K \in \mathbb {N}}\) is bounded in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\). Further, by Lemma 2.13 and since \(\Delta _{h_K} v \rightarrow \partial _t v\) in \(L^1(0,T;L^\Phi (\Omega ))\) as \(K \rightarrow \infty\), we infer

$$\begin{aligned} \lim _{K \rightarrow \infty } \mathrm {I}_K =0. \end{aligned}$$
(3.14)

Next, by the generalized Hölder’s inequality (1.8), Lemma 2.12, the definition of \(\psi ^{(K)}\) and since \(\psi \in C^0([0,T];L^\Phi (\Omega ))\), we deduce that

$$\begin{aligned} |\mathrm {II}_K|&\le \sup _{t \in [0,\tau ]} \Vert b\big (u^{(K)}(t)\big )\Vert '_{L^{\Phi ^*}(\Omega )} \nonumber \\&\quad \cdot \mathop {\int \!\!\!\!\!\!-}\nolimits _{\tau -h_K}^\tau \big [\Vert \psi (\tau )-\psi (t)\Vert _{L^\Phi (\Omega )} +\Vert \psi (\tau )-\psi ^{(K)}(t)\Vert _{L^\Phi (\Omega )}\big ] \,\mathrm {d}t\nonumber \\&\le \sup _{t \in [0,\tau ]} \Vert b\big (u^{(K)}(t)\big )\Vert '_{L^{\Phi ^*}(\Omega )} \nonumber \\&\quad \cdot \left( \mathop {\int \!\!\!\!\!\!-}\nolimits _{\tau -h_K}^\tau \Vert \psi (\tau )-\psi \Vert _{L^\Phi (\Omega )} \,\mathrm {d}t+\sup _{t \in [\tau -h_K,\tau +h_K]} \Vert \psi (\tau )-\psi (t)\Vert _{L^\Phi (\Omega )} \right) \rightarrow 0 \end{aligned}$$
(3.15)

in the limit \(K \rightarrow \infty\). Similarly, we obtain that

$$\begin{aligned} \limsup _{K \rightarrow \infty } |\mathrm {III}_K| \le \lim _{K \rightarrow \infty } \Vert g_o\Vert '_{L^{\Phi ^*}(\Omega )} \mathop {\int \!\!\!\!\!\!-}\nolimits _0^{h_K} \Vert \psi (h_K)-\psi (t)\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t=0. \end{aligned}$$
(3.16)

In order to treat the remaining term in (3.10), we apply the finite integration by parts formula from Lemma 2.14. This yields

$$\begin{aligned} \iint _{\Omega _\tau } \Delta _{-h_K}&b\big (u^{(K)}\big ) \big (v-u^{(K)}\big ) \,\mathrm {d}x\mathrm {d}t\nonumber \\&\le \iint _{\Omega _\tau } \Delta _{h_K} v \big (b(v)-b\big (u^{(K)}\big )\big ) \,\mathrm {d}x\mathrm {d}t \nonumber \\&\quad -\mathrm {B}_\tau (h_K) +\mathrm {B}_0(h_K) +\varvec{\delta }_1(h_K) +\varvec{\delta }_2(h_K), \end{aligned}$$
(3.17)

where we used the abbreviations

$$\begin{aligned} \mathrm {B}_\tau (h_K)&:= \tfrac{1}{h_K} \iint _{\Omega \times (\tau -h_K,\tau )} {\mathfrak {b}}\big [u^{(K)}(t),v(t+h_K)\big ] \,\mathrm {d}x\mathrm {d}t, \\ \mathrm {B}_0(h_K)&:= \tfrac{1}{h_K} \iint _{\Omega \times (-h_K,0)} {\mathfrak {b}}\big [u^{(K)}(t),v(t)\big ] \,\mathrm {d}x\mathrm {d}t= {\mathfrak {B}}[g_o,v(0)]. \end{aligned}$$

Furthermore, the error terms \(\varvec{\delta }_1(h_K)\) and \(\varvec{\delta }_2(h_K)\) are given by

$$\begin{aligned} \varvec{\delta }_1(h_K)&:= \tfrac{1}{h_K} \iint _{\Omega _T} {\mathfrak {b}}[v(t),v(t+h_K)] \,\mathrm {d}x\mathrm {d}t, \\ \varvec{\delta }_2(h_K)&:= \iint _{\Omega \times (-h_K,0)} \Delta _h v \big (b(v(t+h_K))-b\big (u^{(K)}(t)\big )\big ) \,\mathrm {d}x\mathrm {d}t\\&= \iint _{\Omega \times (-h_K,0)} \Delta _h v (b(v(t+h_K))-b(g_o)) \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

For the characterization of \(\mathrm {B}_0(h_K)\) and \(\varvec{\delta }_2(h_K)\), we used that \(u^{(K)}(t) = g_o\) and \(v(t)=v(0)\) for \(t \in (-h_K,0]\). Since \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), Lemma 2.14 implies that

$$\begin{aligned} \lim _{{\mathfrak {K}} \ni K \rightarrow \infty } \varvec{\delta }_1(h_K) = 0 \quad \text {and}\quad \lim _{{\mathfrak {K}} \ni K \rightarrow \infty } \varvec{\delta }_2(h_K) = 0. \end{aligned}$$
(3.18)

Next, we consider the first term on the right-hand side of (3.17). Since we have that \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), we find that \(\Delta _{h_K} v \rightarrow \partial _t v\) strongly in \(L^1(0,T;L^\Phi (\Omega ))\). Further, since \(\big (b\big (u^{(K)}(t)\big )\big )_{K \in \mathbb {N}}\) is bounded in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\) and by (3.8)\(_2\), we know that \(b\big (u^{(K)}\big ) \mathop {\rightharpoondown }\limits ^{*}b(u)\) weakly\(^*\) in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\). Therefore,

$$\begin{aligned} \lim _{{\mathfrak {K}} \ni K \rightarrow \infty } \iint _{\Omega _T} \Delta _{h_K} v \big (b(v)-b\big (u^{(K)}\big )\big ) \,\mathrm {d}x\mathrm {d}t= \iint _{\Omega _T} \partial _t v (b(v)-b(u)) \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$
(3.19)

Since we are not allowed to pass to the limit \({\mathfrak {K}} \ni K \rightarrow \infty\) in \(\mathrm {B}_\tau (h_K)\) directly, we integrate (3.10) over \(\tau \in (t_o,t_o+\delta )\) for some \(\delta \in (0,T)\) and \(t_o \in [0,T-\delta ]\) and divide the result by \(\delta\). Combining this with (3.17), we find that

$$\begin{aligned} \iint _{\Omega _{t_o}}&f\big (Du^{(K)}\big ) \,\mathrm {d}x\mathrm {d}t\nonumber \\&\le \iint _{\Omega _{t_o+\delta }} f(Dv_K) \,\mathrm {d}x\mathrm {d}t+ \mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \iint _{\Omega _\tau } \Delta _{h_K} v \big (b(v)-b\big (u^{(K)}\big )\big ) \,\mathrm {d}x\mathrm {d}t\mathrm {d}\tau \nonumber \\&\quad -\tfrac{1}{\delta } \int _{t_o}^{t_o+\delta -h_K} \int _\Omega {\mathfrak {b}}\big [u^{(K)}(t),v(t+h_K)\big ] \,\mathrm {d}x\mathrm {d}t+ {\mathfrak {B}}[g_o,v(0)] \nonumber \\&\quad +\varvec{\delta }_1(h_K) +\varvec{\delta }_2(h_K) + \iint _{\Omega _T} \Delta _{-h_K} b\big (u^{(K)}\big )\big (\psi ^{(K)}-\psi \big ) \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$
(3.20)

Note that \(u^{(K)} \rightarrow u\) a.e. in \(\Omega _T\). Since \(v \in C^0([0,T];L^\Phi (\Omega ))\), we have that \(v(t+h_K) \rightarrow v(t)\) as \({\mathfrak {K}} \ni K \rightarrow \infty\) a.e. in \(\Omega _T\). Since \({\mathfrak {b}}\) is nonnegative, we may apply Fatou’s lemma, which yields

$$\begin{aligned} \tfrac{1}{\delta } \int _{t_o}^{t_o+\delta } \int _\Omega&{\mathfrak {b}}[u,v] \,\mathrm {d}x\mathrm {d}t\nonumber \\&\le \liminf _{{\mathfrak {K}} \ni K \rightarrow \infty } \tfrac{1}{\delta } \int _{t_o}^{t_o+\delta -h_K} \int _\Omega {\mathfrak {b}}\big [u^{(K)}(t),v(t+h_K)\big ] \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$
(3.21)

Collecting (3.11), (3.12), (3.14), (3.15), (3.16), (3.19) and (3.21), we infer

$$\begin{aligned} \iint _{\Omega _{t_o}} f(Du) \,\mathrm {d}x\mathrm {d}t&\le \iint _{\Omega _{t_o+\delta }} f(Dv) \,\mathrm {d}x\mathrm {d}t+ \mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \iint _{\Omega _\tau } \partial _t v \big (b(v)-b(u)\big ) \,\mathrm {d}x\mathrm {d}t\mathrm {d}\tau \\&\quad +{\mathfrak {B}}[g_o,v(0)] -\tfrac{1}{\delta } \int _{t_o}^{t_o+\delta } \int _\Omega {\mathfrak {b}}[u,v] \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

Now, we pass to the limit \(\delta \downarrow 0\). Observe that the integrals in the first line of the preceding inequality depend continuously on time by the absolute continuity of the integral. For the boundary term, by the fact that \(\Phi (u(t)) \ge 0\), \(b(u)v \ge 0\) and by Lemma 2.3, we find the dominating function

$$\begin{aligned} 0 \le {\mathfrak {b}}[u(t),v(t)] \le \Phi (v(t)) + u(t)b(u(t)) \le \Phi (v(t)) + (m+1) \Phi (u(t)). \end{aligned}$$

Since \(u,v \in C^0([0,T];L^\Phi (\Omega ))\) (see Lemma 2.10 for v), the right-hand side of the preceding inequality depends continuously on time. Therefore, using the dominated convergence theorem, we deduce that

$$\begin{aligned}{}[0,T] \ni t \mapsto \int _\Omega {\mathfrak {b}}[u(t),v(t)] \,\mathrm {d}x\quad \text {is continuous.} \end{aligned}$$

Altogether, passing to the limit \(\delta \downarrow 0\), we conclude that

$$\begin{aligned} \iint _{\Omega _{t_o}} f(Du) \,\mathrm {d}x\mathrm {d}t&\le \iint _{\Omega _{t_o}} f(Dv) \,\mathrm {d}x\mathrm {d}t+ \iint _{\Omega _\tau } \partial _t v \big (b(v)-b(u)\big ) \,\mathrm {d}x\mathrm {d}t\\&\quad +{\mathfrak {B}}[g_o,v(0)] -{\mathfrak {B}}[u(t_o),v(t_o)] \end{aligned}$$

holds true for a.e. \(t_o \in [0,T]\) and any comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). Thus, u is a variational solution to (1.2) in the sense of Definition 1.1. \(\square\)

Existence for less regular data with respect to the spatial variables

Next, we prove an existence result for boundary values and obstacle, whose time derivative is less regular with respect to the spatial variables. More precisely, we assume that \(0 \le \psi \le g\) a.e. on \(\Omega _T\),

$$\begin{aligned} \left\{ \begin{array}{l} g \in L^p(0,T;W^{1,p}(\Omega )), \; \partial _t g \in L^2(0,T;L^\Phi (\Omega )) \cap L^p(0,T;W^{1,p}(\Omega )) ,\\ g_o = g(0) \in L^\Phi (\Omega ) \end{array} \right. \end{aligned}$$
(4.1)

and

$$\begin{aligned} \left\{ \begin{array}{l} \psi \in g + L^p(0,T;W^{1,p}_0(\Omega )), \partial _t \psi \in L^2(0,T;L^\Phi (\Omega )) \cap L^p(0,T;W^{1,p}(\Omega )), \\ \psi (0) \in L^\Phi (\Omega ). \end{array} \right. \end{aligned}$$
(4.2)

For the proof of the desired result, we need the following lemma, cf. [12, Lemma 4.3].

Lemma 4.1

Assume that \(\Omega \subset \mathbb {R}^n\) is a bounded open set with Lipschitz boundary and let \(\Omega _\varepsilon := \{x \in \Omega :{{\,\mathrm{dist}\,}}(x,\partial \Omega ) > \varepsilon \}\) for any \(\varepsilon >0\). Then, for any \(u \in L^p(0,T;W^{1,p}_0(\Omega ))\) we have

$$\begin{aligned} \iint _{(\Omega \setminus \Omega _\varepsilon ) \times (0,T)} |u|^p \,\mathrm {d}x\mathrm {d}t\le c(p,\Omega ) \varepsilon ^p \iint _{(\Omega \setminus \Omega _\varepsilon ) \times (0,T)} |Du|^p \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

Theorem 4.2

Assume that hypotheses (4.1) and (4.2) are satisfies. Then, there exists a variational solution to (1.2) in the sense of Definition 1.1.

The proof of 4.2 is divided into several steps.

Approximation

Let \(\varepsilon >0\), extend g to \(\mathbb {R}^n \times (0,T)\) by zero and define \(g_\varepsilon := g *\eta _\varepsilon + \varepsilon\), where \(\eta _\varepsilon\) denotes the standard mollifier in \(\mathbb {R}^n\). Then, we have that \(g_\varepsilon \in L^p(0,T;W^{1,p}(\Omega ))\), \(g_{o,\varepsilon } := g_\varepsilon (0) \in C^\infty (\mathbb {R}^n) \subset \big (L^2(\Omega ) \cap W^{1,p}(\Omega )\big )\) and

$$\begin{aligned} \partial _t g_\varepsilon = \partial _t g *\eta _\varepsilon \in L^2(0,T;C^\infty (\mathbb {R}^n)) \cap L^p(0,T;W^{1,p}(\Omega )). \end{aligned}$$

In order to define mollifications of the obstacle function and comparison maps, we consider the cutoff function \(\zeta _\varepsilon \in W^{1,\infty }(\Omega ,\mathbb {R}_{\ge 0})\) with \(\zeta _\varepsilon \equiv 0\) in \(\Omega \setminus \Omega _\varepsilon\), \(\zeta _\varepsilon \equiv 1\) in \(\Omega _{2\varepsilon }\) and

$$\begin{aligned} \zeta _\varepsilon (x) := \tfrac{{{\,\mathrm{dist}\,}}(x,\partial \Omega )-\varepsilon }{\varepsilon } \quad \text {for } x \in \Omega _\varepsilon \setminus \Omega _{2\varepsilon }. \end{aligned}$$

Then, we define

$$\begin{aligned} \psi _\varepsilon := (\zeta _\varepsilon \psi + (1-\zeta _\varepsilon )g) *\eta _\varepsilon +\varepsilon . \end{aligned}$$

Note that

$$\begin{aligned} D\psi _\varepsilon = (\zeta _\varepsilon D\psi + (1-\zeta _\varepsilon ) Dg +D\zeta _\varepsilon (\psi -g)) *\eta _\varepsilon . \end{aligned}$$

Since \({{\,\mathrm{spt}\,}}(\zeta _\varepsilon ) = \overline{\Omega _\varepsilon }\), we conclude that \(\psi _\varepsilon \in g_\varepsilon + L^p(0,T;W^{1,p}_0(\Omega ))\) and \(\psi _\varepsilon (0) \in L^2(\Omega ) \cap \big (g_{o,\varepsilon } + W^{1,p}_0(\Omega )\big )\). Further, since \(\zeta _\varepsilon\) and \(\eta _\varepsilon\) are independent of time, we have that

$$\begin{aligned} \partial _t \psi _\varepsilon = (\zeta _\varepsilon \partial _t \psi + (1-\zeta _\varepsilon ) \partial _t g) *\eta _\varepsilon \in L^2(0,T;C^\infty (\Omega )) \cap L^p(\Omega _T) \end{aligned}$$

and

$$\begin{aligned} \partial _t D\psi _\varepsilon = (\zeta _\varepsilon \partial _t D\psi + (1-\zeta _\varepsilon ) \partial _t Dg +D\zeta _\varepsilon (\partial _t\psi - \partial _t g)) *\eta _\varepsilon \in L^p(\Omega _T,\mathbb {R}^n). \end{aligned}$$

Finally, by a standard property of the applied mollification procedure, we find that \(0<\varepsilon \le \psi _\varepsilon \le g_\varepsilon\). More generally, for any comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\), we define the mollification

$$\begin{aligned} v_\varepsilon := (\zeta _\varepsilon v + (1-\zeta _\varepsilon )g) *\eta _\varepsilon +\varepsilon . \end{aligned}$$

Then, we obtain that \(v_\varepsilon \in g_\varepsilon + L^p(0,T;W^{1,p}_0(\Omega ))\), \(\partial _t v_\varepsilon \in L^1(0,T;L^\Phi (\Omega ))\), \(v_\varepsilon (0) \in L^\Phi (\Omega )\) and \(v_\varepsilon \ge \psi _\varepsilon\) a.e. in \(\Omega _T\). Next, we prove that

$$\begin{aligned} v_\varepsilon \rightarrow v \text{ in } L^\Phi (\Omega _T) \text{ as } \varepsilon \downarrow 0. \end{aligned}$$
(4.3)

Indeed, we conclude that

$$\begin{aligned} \Vert v_\varepsilon - v\Vert _{L^\Phi (\Omega _T)}&\le \Vert v_\varepsilon - v *\eta _\varepsilon \Vert _{L^\Phi (\Omega _T)} +\Vert v *\eta _\varepsilon - v\Vert _{L^\Phi (\Omega _T)} \\&= \Vert ((1-\zeta _\varepsilon )(g-v)) *\eta _\varepsilon + \varepsilon \Vert _{L^\Phi (\Omega _T)} +\Vert v *\eta _\varepsilon - v\Vert _{L^\Phi (\Omega _T)} \\&\le \Vert (1-\zeta _\varepsilon )(g-v)\Vert _{L^\Phi (\Omega _T)} +\Vert \varepsilon \Vert _{L^\Phi (\Omega _T)} +\Vert v *\eta _\varepsilon - v\Vert _{L^\Phi (\Omega _T)} \rightarrow 0 \end{aligned}$$

in the limit \(\varepsilon \downarrow 0\). Further, (4.3) in particular implies that there exists a (not relabeled) subsequence such that \(v_\varepsilon \rightarrow v\) a.e in \(\Omega _T\). A similar computation shows that

$$\begin{aligned} v_\varepsilon (0) \rightarrow v(0) \text{ in } L^\Phi (\Omega ) \text{ and } \text{ a.e } \text{ in } \Omega \text{ as } \varepsilon \downarrow 0. \end{aligned}$$
(4.4)

Moreover, observe that

$$\begin{aligned} \sup _{t \in [0,T]} \Vert v_\varepsilon (t)\Vert _{L^\Phi (\Omega )}&\le \sup _{t \in [0,T]} \Vert \zeta _\varepsilon v(t) + (1-\zeta _\varepsilon )g(t)\Vert _{L^\Phi (\Omega )} +\Vert \varepsilon \Vert _{L^\Phi (\Omega )} \nonumber \\&\le \sup _{t \in [0,T]} \Vert v(t)\Vert _{L^\Phi (\Omega )} +\sup _{t \in [0,T]} \Vert g(t)\Vert _{L^\Phi (\Omega )} +\Vert 1\Vert _{L^\Phi (\Omega )}. \end{aligned}$$
(4.5)

Hence, by Remark 2.4 the sequence \((b(v_\varepsilon ))_{\varepsilon >0}\) is bounded in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\). Together, the preceding considerations prove that there exists another subsequence such that

$$\begin{aligned} b(v_\varepsilon ) \mathop {\rightharpoondown }\limits ^{*}b(v) \text{ weakly }^* \text{ in } L^\infty (0,T;L^{\Phi ^*}(\Omega )) \text{ as } \varepsilon \downarrow 0. \end{aligned}$$
(4.6)

Next, we compute that

$$\begin{aligned} \Vert \partial _t v_\varepsilon&- \partial _t v\Vert _{L^1(0,T;L^\Phi (\Omega ))} \\&\le \Vert \partial _t v_\varepsilon - \partial _t v *\eta _\varepsilon \Vert _{L^1(0,T;L^\Phi (\Omega ))} +\Vert \partial _t v *\eta _\varepsilon - \partial _t v \Vert _{L^1(0,T;L^\Phi (\Omega ))} \\&\le \Vert (1-\zeta _\varepsilon )(\partial _t g - \partial _t v) \Vert _{L^1(0,T;L^\Phi (\Omega ))} +\Vert \partial _t v *\eta _\varepsilon - \partial _t v \Vert _{L^1(0,T;L^\Phi (\Omega ))} \rightarrow 0 \end{aligned}$$

in the limit \(\varepsilon \downarrow 0\), which yields

$$\begin{aligned} \partial_t v_\varepsilon \rightarrow \partial _t v \text{ in } L^1(0,T;L^\Phi (\Omega )) \text{ as } \varepsilon \downarrow 0. \end{aligned}$$
(4.7)

Finally, we show that

$$\begin{aligned} Dv_\varepsilon \rightarrow Dv \text{ in } L^p(\Omega _T,\mathbb {R}^n) \text{ as } \varepsilon \downarrow 0. \end{aligned}$$
(4.8)

To this end, we first compute

$$\begin{aligned} \Vert Dv_\varepsilon - Dv\Vert _{L^p(\Omega _T,\mathbb {R}^n)}&\le \Vert Dv_\varepsilon - Dv *\eta _\varepsilon \Vert _{L^p(\Omega _T,\mathbb {R}^n)} +\Vert Dv *\eta _\varepsilon - Dv\Vert _{L^p(\Omega _T,\mathbb {R}^n)} \\&\le \Vert (1-\zeta _\varepsilon )(Dg-Dv) +D\zeta _\varepsilon (v-g)\Vert _{L^p(\Omega _T,\mathbb {R}^n)} \\&\quad +\Vert Dv *\eta _\varepsilon - Dv\Vert _{L^p(\Omega _T,\mathbb {R}^n)}. \end{aligned}$$

The second term on the right-hand side of the preceding inequality clearly vanishes as \(\varepsilon \downarrow 0\). For the first term, by definition of \(\zeta _\varepsilon\) and Lemma 4.1 we conclude that

$$\begin{aligned} \iint _{\Omega _T}&|(1-\zeta _\varepsilon )(Dg-Dv)+D\zeta _\varepsilon (v-g)|^p \,\mathrm {d}x\mathrm {d}t\\&\le 2^{p-1} \iint _{\Omega _T} \big [(1-\zeta _\varepsilon )^p|Dg-Dv|^p + |D\zeta _\varepsilon |^p |v-g|^p\big ] \,\mathrm {d}x\mathrm {d}t\\&\le 2^{p-1} \iint _{(\Omega \setminus \Omega _{2\varepsilon }) \times (0,T)} \big [|Dg-Dv|^p + \varepsilon ^{-p}|v-g|^p\big ] \,\mathrm {d}x\mathrm {d}t\\&\le c(n,p,\Omega ) \iint _{(\Omega \setminus \Omega _{2\varepsilon }) \times (0,T)} |Dg-Dv|^p \,\mathrm {d}x\mathrm {d}t\rightarrow 0 \end{aligned}$$

in the limit \(\varepsilon \downarrow 0\). This proves (4.8).

Solutions of the regularized problem

For any \(\varepsilon >0\), the mollifications \(g_\varepsilon\) and \(\psi _\varepsilon\) satisfy the assumptions of Theorem 3.1. Hence, there exist corresponding variational solutions \(u_\varepsilon \in C^0([0,T];L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}_0(\Omega )) \big)\). Applying Lemma 2.20 with the admissible comparison map \(g_\varepsilon\), we obtain the energy bound

$$\begin{aligned} \tfrac{1}{2} \sup _{t \in [0,T]} \varrho _\Omega (u_\varepsilon (t)) + \iint _{\Omega _T} f(Du_\varepsilon ) \,\mathrm {d}x\le C, \end{aligned}$$
(4.9)

where the constant C is defined by

$$\begin{aligned} C&:= \tfrac{2^\ell }{m} +\tfrac{1}{2} \sup _{\varepsilon>0} \sup _{t \in [0,T]} \varrho _\Omega (g_\varepsilon (t)) +c(\ell ,m)\sup _{\varepsilon>0} \bigg (\int _0^T \Vert \partial _t g_\varepsilon \Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\bigg )^\frac{\ell +1}{\ell } \\&\quad +\tfrac{3}{\ell } \sup _{\varepsilon >0} \iint _{\Omega _T} f(Dg_\varepsilon ) \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

By (4.5) and (1.7), (4.7) and (4.8) together with the growth condition (1.3) C is finite. Therefore, there exists a (not relabeled) subsequence and a limit map \(u \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}_0(\Omega )) \big)\) such that

$$\begin{aligned} \left\{ \begin{array}{l} u_\varepsilon \mathop {\rightharpoondown }\limits ^{*}u \text{ weakly}^* \text{ in } L^\infty (0,T;L^\Phi (\Omega )), \\ \hbox {} u_\varepsilon \rightharpoondown u \hbox { weakly in} \, L^p(0,T;W^{1,p}(\Omega )) \end{array} \right. \end{aligned}$$
(4.10)

as \(\varepsilon \downarrow 0\). By (4.3) applied to \(v=\psi\) and (4.10)\(_2\), we obtain that

$$\begin{aligned} \iint _{\Omega _T} (u-\psi ) \varphi \,\mathrm {d}x= \lim _{\varepsilon \downarrow 0} \iint _{\Omega _T} (u_\varepsilon -\psi _\varepsilon ) \varphi \,\mathrm {d}x\ge 0 \qquad \forall \varphi \in C^\infty _0(\Omega _T,\mathbb {R}_{\ge 0}). \end{aligned}$$

Hence, the obstacle condition \(u \ge \psi\) is satisfied.

Improved convergence of the solutions

Next, we need to establish

$$\begin{aligned} b(u_\varepsilon ) \mathop {\rightharpoondown }\limits ^{*}b(u) \text{ weakly }^* \text{ in } L^\infty (0,T;L^{\Phi ^*}(\Omega )) \text{ as } \varepsilon \downarrow 0. \end{aligned}$$
(4.11)

Since \((b(u_\varepsilon ))_\varepsilon\) is bounded in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\) by (4.10) and Remark 2.4, there exists a subsequence such that

$$\begin{aligned} b(u_\varepsilon ) \mathop {\rightharpoondown }\limits ^{*}w \text{ weakly }^* \text{ in } L^\infty (0,T;L^{\Phi ^*}(\Omega )) \text{ as } \varepsilon \downarrow 0 \end{aligned}$$
(4.12)

for some limit map \(w \in L^\infty (0,T;L^{\Phi ^*}(\Omega ))\). Therefore, it remains to prove that w has the structure b(u). To this end, for \(h>0\) we consider mollifications \([u_\varepsilon -\psi _\varepsilon ]_h\) and \([u-\psi ]_h\) according to (2.7) with zero initial values and define

$$\begin{aligned} w_{\varepsilon ,h} := [u_\varepsilon -\psi _\varepsilon ]_h + \psi _\varepsilon \text{ and } w_h := [u-\psi ]_h + \psi . \end{aligned}$$

Since \(L^\Phi (\Omega )\) is separable, by Lemma 2.15 we obtain that \(w_{\varepsilon ,h}, w_h \in L^\infty (0,T;L^\Phi (\Omega ))\). Further, we find that \(w_{\varepsilon ,h} \in g_\varepsilon + L^p(0,T;W^{1,p}_0(\Omega ))\) and \(w_h \in g + L^p(0,T;W^{1,p}_0(\Omega ))\). Since \(u_\varepsilon \ge \psi _\varepsilon\), we have that \(w_{\varepsilon ,h} \ge \psi _\varepsilon\). Next, note that (2.6) implies

$$\begin{aligned} \partial _t [u_\varepsilon -\psi _\varepsilon ]_h = \tfrac{1}{h} \big ((u_\varepsilon -\psi _\varepsilon ) - [u_\varepsilon -\psi _\varepsilon ]_h\big ) \in L^\Phi (\Omega _T) \cap L^p(\Omega _T). \end{aligned}$$

Thus, by (4.10)\(_2\) and since \(\psi _\varepsilon \rightarrow \psi\) in \(L^p(0,T;W^{1,p}(\Omega ))\), we deduce that for any fixed \(h>0\) the sequence \((\partial _t [u_\varepsilon -\psi _\varepsilon ]_h)_\varepsilon\) is bounded in \(L^p(\Omega _T)\). Further, by Lemma 2.17

$$\begin{aligned} w_{\varepsilon ,h} \rightharpoondown w_h \text{ weakly } \text{ in } L^p(0,T;W^{1,p}(\Omega )) \text{ as } \varepsilon \downarrow 0 \end{aligned}$$
(4.13)

holds true for fixed \(h>0\). Therefore, from Rellich’s theorem, we infer

$$\begin{aligned} w_{\varepsilon ,h} \rightarrow w_h \text{ in } L^p(\Omega _T) \text{ as } \varepsilon \downarrow 0. \end{aligned}$$
(4.14)

We did not have to pass to a subsequence, since the limit is determined by (4.13). Next, we use \(w_{\varepsilon ,h}\) as comparison map in the variational inequality satisfied by \(u_\varepsilon\). Discarding the nonnegative terms \(\iint _{\Omega _T} f(Du_\varepsilon ) \,\mathrm {d}x\mathrm {d}t\) and \({\mathfrak {B}}[u_\varepsilon (T),w_{\varepsilon ,h}(T)]\), we deduce that

$$\begin{aligned}&-\iint _{\Omega _T} \partial _t w_{\varepsilon ,h} \big (b(w_{\varepsilon ,h}) - b(u_\varepsilon )\big ) \,\mathrm {d}x\mathrm {d}t\\&\quad = \tfrac{1}{h} \iint _{\Omega _T} (w_{\varepsilon ,h}-u_\varepsilon )\big (b(w_{\varepsilon ,h}) - b(u_\varepsilon )\big ) \,\mathrm {d}x\mathrm {d}t-\iint _{\Omega _T} \partial _t \psi _\varepsilon \big (b(w_{\varepsilon ,h}) - b(u_\varepsilon )\big ) \,\mathrm {d}x\mathrm {d}t\\&\quad \le \iint _{\Omega _T} f(Dw_{\varepsilon ,h}) \,\mathrm {d}x\mathrm {d}t+{\mathfrak {B}}[g_{o,\varepsilon },\psi _\varepsilon (0)], \end{aligned}$$

which is equivalent to

$$\begin{aligned}&\iint _{\Omega _T} (w_{\varepsilon ,h}-u_\varepsilon )\big (b(w_{\varepsilon ,h}) - b(u_\varepsilon )\big ) \,\mathrm {d}x\mathrm {d}t\nonumber \\&\quad \le h\iint _{\Omega _T} \partial _t \psi _\varepsilon \big (b(w_{\varepsilon ,h})- b(u_\varepsilon )\big ) \,\mathrm {d}x\mathrm {d}t+h\iint _{\Omega _T} f(Dw_{\varepsilon ,h}) \,\mathrm {d}x\mathrm {d}t\nonumber \\&\qquad +h{\mathfrak {B}}[g_{o,\varepsilon },\psi _\varepsilon (0)] \nonumber \\&\quad =: h\mathrm {I}_h + h\mathrm {II}_h + h\mathrm {III}, \end{aligned}$$
(4.15)

where the definition of \(\mathrm {I}_h\), \(\mathrm {II}_h\) and \(\mathrm {III}\) is clear in this context. By the generalized Hölder’s inequality (1.8), Hölder’s inequality, Lemmas 2.3 and 2.5, we infer

$$\begin{aligned} |\mathrm {I}_h|&\le \Vert \partial _t \psi _\varepsilon \Vert _{L^1(0,T;L^\Phi (\Omega ))} \big [\sup _{t \in [0,T]} \Vert b(w_{\varepsilon ,h}(t))\Vert '_{L^{\Phi ^*}(\Omega )} +\sup _{t \in [0,T]} \Vert b(u_\varepsilon (t))\Vert '_{L^{\Phi ^*}(\Omega )}\big ] \\&\le \Vert \partial _t \psi _\varepsilon \Vert _{L^1(0,T;L^\Phi (\Omega ))} \big [2+ \sup _{t \in [0,T]} \varrho _\Omega ^*(b(w_{\varepsilon ,h}(t)))^\frac{1}{\ell +1} +\sup _{t \in [0,T]} \varrho _\Omega ^*(b(u_\varepsilon (t)))^\frac{1}{\ell +1}\big ] \\&\le \Vert \partial _t \psi _\varepsilon \Vert _{L^1(0,T;L^\Phi (\Omega ))} \big [2+ \sup _{t \in [0,T]} (m \varrho _\Omega (w_{\varepsilon ,h}(t)))^\frac{1}{\ell +1} +\sup _{t \in [0,T]} (m \varrho _\Omega (u_\varepsilon (t)))^\frac{1}{\ell +1}\big ]. \end{aligned}$$

Note that \(\sup _{t \in [0,T]} \varrho _\Omega (u_\varepsilon (t)) \le 2C\) and \((\Vert \partial _t \psi _\varepsilon \Vert _{L^1(0,T;L^\Phi (\Omega ))})_\varepsilon\) is bounded by (4.7). Further, by Lemma 2.15, (4.5) and (4.10), we find that

$$\begin{aligned} \sup _{\varepsilon>0} \Vert w_{\varepsilon ,h}\Vert _{L^\infty (0,T;L^\Phi (\Omega ))}&\le \sup _{\varepsilon>0} \Vert [u_\varepsilon -\psi _\varepsilon ]_h\Vert _{L^\infty (0,T;L^\Phi (\Omega ))} +\sup _{\varepsilon>0} \Vert \psi _\varepsilon \Vert _{L^\infty (0,T;L^\Phi (\Omega ))} \\&\le \sup _{\varepsilon>0} \Vert u_\varepsilon -\psi _\varepsilon \Vert _{L^\infty (0,T;L^\Phi (\Omega ))} +\sup _{\varepsilon >0} \Vert \psi _\varepsilon \Vert _{L^\infty (0,T;L^\Phi (\Omega ))} \\&<\infty . \end{aligned}$$

Thus, taking into account (1.7), we conclude that

$$\begin{aligned} |\mathrm {I}_h| \le c \end{aligned}$$
(4.16)

for a constant c independent of \(\varepsilon\) and h. Next, by the growth condition (1.3), Lemma 2.15, (4.8) and (4.9), we obtain that

$$\begin{aligned} |\mathrm {II}_h|&\le L\iint _{\Omega _t} \big [1 + |Dw_{\varepsilon ,h}|^p\big ] \,\mathrm {d}x\mathrm {d}t\nonumber \\&\le L|\Omega |T +2^{p-1}L\big [\Vert [Du_\varepsilon -D\psi _\varepsilon ]_h\Vert _{L^p(\Omega _T,\mathbb {R}^n)}^p +\Vert D\psi _\varepsilon \Vert _{L^p(\Omega _t,\mathbb {R}^n)}^p\big ] \nonumber \\&\le L|\Omega |T +c(p)L\big [\Vert Du_\varepsilon \Vert _{L^p(\Omega _T,\mathbb {R}^n)}^p +\Vert D\psi _\varepsilon \Vert _{L^p(\Omega _t,\mathbb {R}^n)}^p\big ] \nonumber \\&\le L|\Omega |T +c(p)L\big [C +\sup _{\varepsilon >0} \Vert D\psi _\varepsilon \Vert _{L^p(\Omega _T,\mathbb {R}^n)}^p\big ] <\infty . \end{aligned}$$
(4.17)

Finally, by (4.4) applied to \(\psi (0)\) and \(g_o\), we have that \({\mathfrak {b}}[g_{o,\varepsilon },\psi _\varepsilon (0)] \rightarrow {\mathfrak {b}}[g_o,\psi (0)]\) a.e. in \(\Omega _T\) as \(\varepsilon \downarrow 0\). Further, by Lemma 2.3 and since \(b(g_{o,\varepsilon }),\psi _\varepsilon (0) \ge 0\) by Lemma 2.11, we find the dominating function

$$\begin{aligned} 0&\le {\mathfrak {b}}[g_{o,\varepsilon },\psi _\varepsilon (0)] \le \Phi (\psi _\varepsilon (0)) + \Phi ^*(b(g_{o,\varepsilon })) \le \Phi (\psi _\varepsilon (0)) + m\Phi (g_{o,\varepsilon }) \\&\rightarrow \Phi (\psi (0)) + m\Phi (g_o) \end{aligned}$$

in \(L^1(\Omega )\) as \(\varepsilon \downarrow 0\). Thus, we conclude that \({\mathfrak {B}}[g_{o,\varepsilon },\psi _\varepsilon (0)] \rightarrow {\mathfrak {B}}[g_o,\psi (0)]\) in the limit \(\varepsilon \downarrow 0\). In particular, this implies

$$\begin{aligned} \sup _{\varepsilon >0} \mathrm {III} <\infty . \end{aligned}$$
(4.18)

Inserting (4.16), (4.17) and (4.18) into (4.15), we infer

$$\begin{aligned} \iint _{\Omega _T} (w_{\varepsilon ,h}-u_\varepsilon )\big (b(w_{\varepsilon ,h}) - b(u_\varepsilon )\big ) \,\mathrm {d}x\mathrm {d}t\le ch, \end{aligned}$$

where the constant c is independent of \(\varepsilon\) and h. By Lemma 2.9, Hölder’s inequality and Lemma 2.7, we bound the left-hand side of the preceding inequality from below,

$$\begin{aligned}&\varrho _{\Omega _T}(w_{\varepsilon ,h} - u_\varepsilon ) \le \iint _{\Omega _T} |\Phi (w_{\varepsilon ,h}) - \Phi (u_\varepsilon )| \,\mathrm {d}x\mathrm {d}t\\&\quad \le \bigg (\iint _{\Omega _T} \big |\sqrt{\Phi (w_{\varepsilon ,h})} + \sqrt{\Phi (u_\varepsilon )}\big |^2 \,\mathrm {d}x\mathrm {d}t\bigg )^\frac{1}{2} \bigg (\iint _{\Omega _T} \big |\sqrt{\Phi (w_{\varepsilon ,h})} - \sqrt{\Phi (u_\varepsilon )}\big |^2 \,\mathrm {d}x\mathrm {d}t\bigg )^\frac{1}{2} \\&\quad \le c \big [ \varrho _{\Omega _T}(w_{\varepsilon ,h})^\frac{1}{2} + \varrho _{\Omega _T}(u_\varepsilon )^\frac{1}{2} \big ] \bigg (\iint _{\Omega _T} (w_{\varepsilon ,h}-u_\varepsilon )\big (b(w_{\varepsilon ,h}) - b(u_\varepsilon )\big ) \,\mathrm {d}x\mathrm {d}t\bigg )^\frac{1}{2} \end{aligned}$$

with a constant \(c=c(\ell ,m)\). Since we have already shown that \(\iint _{\Omega _T} \Phi (w_{\varepsilon ,h}) \,\mathrm {d}x\mathrm {d}t\) and \(\iint _{\Omega _T} \Phi (u_\varepsilon ) \,\mathrm {d}x\mathrm {d}t\) stay bounded in the limits \(\varepsilon \downarrow 0\) and \(h \downarrow 0\), we deduce that

$$\begin{aligned} \varrho _{\Omega _T}(w_{\varepsilon ,h} - u_\varepsilon ) \le ch^\frac{1}{2}, \end{aligned}$$

where the constant c does not depend on \(\varepsilon\) or h. Further, by (1.6) and Lemma 2.8 for any \(\delta >0\), there exists \(h_o>0\) such that

$$\begin{aligned} \Vert w_{\varepsilon ,h} - u_\varepsilon \Vert _{L^{\ell +1}(\Omega _T)} <\tfrac{\delta }{2} \end{aligned}$$
(4.19)

holds true for any \(0<h<h_o\) and any \(\varepsilon >0\). Choosing \(h_o\) smaller if necessary, by Lemma 2.15 we may assume that

$$\begin{aligned} \Vert w_h - u\Vert _{L^{\ell +1}(\Omega _T)} <\tfrac{\delta }{2}. \end{aligned}$$
(4.20)

Let \(q := \min \{\ell +1,p\}\). Then, by (4.19) and (4.20) with a suitable choice of \(h>0\) and (4.14), we infer

$$\begin{aligned} \lim _{\varepsilon \downarrow 0}&\Vert u-u_\varepsilon \Vert _{L^q(\Omega _T)} \\&\le \lim _{\varepsilon \downarrow 0} \big [\Vert u-w_h\Vert _{L^q(\Omega _T)} +\Vert w_h-w_{\varepsilon ,h}\Vert _{L^q(\Omega _T)} +\Vert w_{\varepsilon ,h}-u_\varepsilon \Vert _{L^q(\Omega _T)}\big ] \\&< \delta +\lim _{\varepsilon \downarrow 0} \Vert w_h-w_{\varepsilon ,h}\Vert _{L^q(\Omega _T)} =\delta . \end{aligned}$$

Since \(\delta\) was arbitrary, this leads to

$$\begin{aligned} u_\varepsilon \rightarrow u \text{ in } L^{\min \{\ell +1,p\}}(\Omega _T) \text{ as } \varepsilon \downarrow 0. \end{aligned}$$

Passing to a (not relabeled) subsequence, we have that \(u_\varepsilon \rightarrow u\) a.e. in \(\Omega _T\) as \(\varepsilon \downarrow 0\). Together with (4.12), this implies (4.11).

Passage to the limit

Consider a comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). Define mollifications \(v_\varepsilon\) as in Sect. 4.1. Because of the boundary term, we are not allowed to pass to the limit \(\varepsilon \downarrow 0\) in the variational inequality satisfied by \(u_\varepsilon\) and \(v_\varepsilon\) directly. Instead, we integrate over \(\tau \in (t_o,t_o+\delta )\) for some \(\delta \in (0,T)\) and \(t_o \in (0,T-\delta ]\). This leads to

$$\begin{aligned} \iint _{\Omega _{t_o}}&f(Du_\varepsilon ) \,\mathrm {d}x\mathrm {d}t+\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } {\mathfrak {B}}[u_\varepsilon (\tau ),v_\varepsilon (\tau )] \,\mathrm {d}\tau \nonumber \\&\le \iint _{\Omega _{t_o+\delta }} f(Dv_\varepsilon ) \,\mathrm {d}x\mathrm {d}t+\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \iint _{\Omega _\tau } \partial _t v_\varepsilon \big (b(v_\varepsilon ) - b(u_\varepsilon )\big ) \,\mathrm {d}x\mathrm {d}t\mathrm {d}\tau \nonumber \\&\quad +{\mathfrak {B}}[g_{o,\varepsilon },v_\varepsilon (0)]. \end{aligned}$$
(4.21)

Since f is convex and satisfies the coercivity condition (1.3)\(_1\), by (4.10)\(_2\) we have that

$$\begin{aligned} \iint _{\Omega _{t_o}} f(Du) \,\mathrm {d}x\mathrm {d}t\le \liminf _{\varepsilon \downarrow 0} \iint _{\Omega _{t_o}} f(Du_\varepsilon ) \,dx\mathrm {d}t. \end{aligned}$$
(4.22)

Further, note that

$$\begin{aligned} {\mathfrak {B}}[u_\varepsilon (\tau ),v_\varepsilon (\tau )] = \int _\Omega \big [\Phi (v_\varepsilon (\tau )) + \Phi ^*(b(u_\varepsilon (\tau ))) -b(u_\varepsilon (\tau ))v_\varepsilon (\tau )\big ] \,\mathrm {d}x. \end{aligned}$$

By (1.6), the functionals \(\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \int _\Omega \Phi (\cdot ) \,\mathrm {d}x\mathrm {d}\tau\) and \(\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \int _\Omega \Phi ^*(\cdot ) \,\mathrm {d}x\mathrm {d}\tau\) are continuous with respect to strong convergence in \(L^\Phi (\Omega _T)\), respectively \(L^{\Phi ^*}(\Omega _T)\). Together with the convexity of \(\Phi\) and \(\Phi ^*\), this implies that they are lower semicontinuous with respect to weak convergence in \(L^\Phi (\Omega _T)\), respectively \(L^{\Phi ^*}(\Omega _T)\), cf. [14, Corollary 3.9]. Therefore, by (4.3) and (4.11), we conclude that

$$\begin{aligned} \mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } {\mathfrak {B}}[u(\tau ),v(\tau )] \,\mathrm {d}\tau \le \liminf _{\varepsilon \downarrow 0} \mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } {\mathfrak {B}}[u_\varepsilon (\tau ),v_\varepsilon (\tau )] \,\mathrm {d}\tau . \end{aligned}$$
(4.23)

Next, by the local Lipschitz condition (1.4) and (4.8), we find that

$$\begin{aligned} \lim _{\varepsilon \downarrow 0} \iint _{\Omega _{t_o+\delta }} f(Dv_\varepsilon ) \,\mathrm {d}x\mathrm {d}t= \iint _{\Omega _{t_o+\delta }} f(Dv) \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$
(4.24)

From (4.6), (4.7), (4.11) and the dominated convergence theorem, we infer

$$\begin{aligned} \lim _{\varepsilon \downarrow 0} \mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \iint _{\Omega _\tau }&\partial _t v_\varepsilon \big (b(v_\varepsilon ) - b(u_\varepsilon )\big ) \,\mathrm {d}x\mathrm {d}t\mathrm {d}\tau \nonumber \\&= \mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \iint _{\Omega _\tau } \partial _t v \big (b(v) - b(u)\big ) \,\mathrm {d}x\mathrm {d}t\mathrm {d}\tau . \end{aligned}$$
(4.25)

Finally, by (4.4) applied to v and g and the dominated convergence theorem, we have that

$$\begin{aligned} \lim _{\varepsilon \downarrow 0} {\mathfrak {B}}[g_{o,\varepsilon },v_\varepsilon (0)] = {\mathfrak {B}}[g_o,v(0)]. \end{aligned}$$
(4.26)

Inserting (4.22), (4.23), (4.24), (4.25) and (4.26) into (4.21), we deduce that

$$\begin{aligned} \iint _{\Omega _{t_o}}&f(Du) \,\mathrm {d}x\mathrm {d}t+\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } {\mathfrak {B}}[u(\tau ),v(\tau )] \,\mathrm {d}\tau \\&\le \iint _{\Omega _{t_o+\delta }} f(Dv) \,\mathrm {d}x\mathrm {d}t+\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \iint _{\Omega _\tau } \partial _t v \big (b(v) - b(u)\big ) \,\mathrm {d}x\mathrm {d}t\mathrm {d}\tau +{\mathfrak {B}}[g_o,v(0)] \end{aligned}$$

holds true for any \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). Passing to the limit \(\delta \downarrow 0\), we conclude that u is a variational solution to (1.2) in the sense of Definition 1.1. \(\square\)

Proof of Theorem 1.2

Regularization

We consider a sequence \(0< h_i \downarrow 0\) as \(i \rightarrow \infty\) and set \(\varepsilon _i := h_i^\frac{1}{2(n+1)p}\). Then, we extend the initial datum \(g_o\) by zero to a function \(g_o \in L^\Phi (\mathbb {R}^n)\) and define mollifications of the extension by

$$\begin{aligned} g_{o,i} := g_o *\eta _{\varepsilon _i}, \end{aligned}$$

where \(\eta _\varepsilon\) denotes a standard mollifier in \(\mathbb {R}^n\). Thus, we have that \(g_{o,i} \in L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\). Further, we set

$$\begin{aligned} g_i := [g]_{h_i}, \end{aligned}$$

where \([g]_{h_i}\) denotes the mollification in time according to (2.7) with initial values \(g_{o,i}\). Since the space \(L^\Phi (\Omega )\) is separable, Lemma 2.15 implies \(g_i \in C^0([0,T];L^\Phi (\Omega )) \cap L^p(0,T;W^{1,p}(\Omega ))\). Moreover, by (2.6) we obtain

$$\begin{aligned} \partial _t g_i = \tfrac{1}{h_i}(g - g_i) \in L^\infty (0,T;L^\Phi (\Omega )) \cap L^p(0,T;W^{1,p}(\Omega )). \end{aligned}$$

Next, we define mollifications of comparison maps \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). To this end, we set

$$\begin{aligned} v_{o,i} := (\chi _{\Omega \setminus \Omega _{\varepsilon _i}} g_o +\chi _{\Omega _{\varepsilon _i}} v(0)) *\eta _{\varepsilon _i}, \end{aligned}$$

where \(\chi\) is the indicator function, i.e., for \(A \subset \mathbb {R}^n\) we have that \(\chi _A \equiv 1\) in A and \(\chi _A \equiv 0\) in \(\mathbb {R}^n \setminus A\). By definition of \(\eta _\varepsilon\), we obtain that \(v_{o,i} \in L^\Phi (\Omega ) \cap \big (g_{o,i} + W^{1,p}_0(\Omega )\big )\). Then, by

$$\begin{aligned} v_i := [v]_{h_i} \end{aligned}$$

we denote the mollification given by (2.7) with initial datum \(v_{o,i}\). Lemma 2.15 implies that \(v_i \in C^0([0,T];L^\Phi (\Omega )) \cap \big (g_i + L^p(0,T;W^{1,p}_0(\Omega ))\big )\). Moreover, by (2.6), we conclude that

$$\begin{aligned} \partial _t v_i = \tfrac{1}{h_i}(v - v_i) \in L^\infty (0,T;L^\Phi (\Omega )) \cap L^p(0,T;W^{1,p}(\Omega )). \end{aligned}$$

In particular, we apply the preceding mollification procedure to \(v=\psi\). Since \(\psi _{o,i} \le g_{o,i}\) by Lemma 2.11 and the definition of \(\eta _\varepsilon\), we have that \(0 \le \psi _i \le g_i\). In the following, we will also use the mollification

$$\begin{aligned} {\tilde{v}}_i := [v]_{h_i} \end{aligned}$$

according to (2.7) with \(v_o = v(0)\). In particular, we know that \({\tilde{v}}_i = v_i + e^{-\frac{t}{h_i}}(v(0)-v_{o,i})\). First, we find that

$$\begin{aligned} v_{o,i} \rightarrow v(0) \text{ a.e. } \text{ and } \text{ in } L^\Phi (\Omega ) \text{ as } i \rightarrow \infty . \end{aligned}$$
(5.1)

Then, we claim that

$$\begin{aligned} v_i \rightarrow v \text{ in } L^\infty (0,T;L^\Phi (\Omega )) \text{ as } i \rightarrow \infty . \end{aligned}$$
(5.2)

Indeed, from Lemma 2.15, we conclude that \({\tilde{v}}_i \rightarrow v\) in \(L^\infty (0,T;L^\Phi (\Omega ))\) as \(i \rightarrow \infty\). Together with (5.1), we obtain that

$$\begin{aligned} \lim _{i \rightarrow \infty }&\Vert v_i-v\Vert _{L^\infty (0,T;L^\Phi (\Omega ))} \\&\le \lim _{i \rightarrow \infty }\Vert v-{\tilde{v}}_i\Vert _{L^\infty (0,T;L^\Phi (\Omega ))} +\lim _{i \rightarrow \infty } \Vert e^{-\frac{t}{h_i}}(v(0)-v_{o,i})\Vert _{L^\infty (0,T;L^\Phi (\Omega ))} \\&\le \lim _{i \rightarrow \infty }\Vert {\tilde{v}}_i-v\Vert _{L^\infty (0,T;L^\Phi (\Omega ))} +\lim _{i \rightarrow \infty }\Vert v_{o,i}-v(0)\Vert _{L^\Phi (\Omega )} =0. \end{aligned}$$

Moreover, from Lemma 2.3, (1.7) and (5.2), we conclude that

$$\begin{aligned} \sup _{t \in [0,T]} \varrho _\Omega ^*(b(v_i(t))) \le \sup _{t \in [0,T]} m \varrho _\Omega (v_i(t)) <\infty . \end{aligned}$$

Together with (5.2), this implies that

$$\begin{aligned} b(v_i) \mathop {\rightharpoondown }\limits ^{*}b(v) \text{ weakly }^* \text{ in } L^\infty (0,T;L^{\Phi ^*}(\Omega )) \text{ as } i \rightarrow \infty . \end{aligned}$$
(5.3)

Next, we show that

$$\begin{aligned} \partial _t v_i \rightarrow \partial _t v \text{ in } L^1(0,T;L^\Phi (\Omega )) \text{ as } i \rightarrow \infty . \end{aligned}$$
(5.4)

To this end, from Lemma 2.16, we infer

$$\begin{aligned} \partial _t {\tilde{v}}_i(t) = \tfrac{1}{h_i} \int _0^t e^\frac{s-t}{h_i} \partial _t v(s) \,\mathrm {d}s, \end{aligned}$$

i.e., the mollification of \(\partial _t v\) according to (2.7) with zero initial datum. Therefore, by Lemma 2.15, we obtain that \(\partial _t {\tilde{v}}_i \rightarrow \partial _t v\) in \(L^1(0,T;L^\Phi (\Omega ))\) as \(i \rightarrow \infty\), which allows us to compute

$$\begin{aligned}&\lim _{i \rightarrow \infty } \Vert \partial _t v_i - \partial _t v\Vert _{L^1(0,T;L^\Phi (\Omega ))} \\&\quad \le \lim _{i \rightarrow \infty } \Vert \partial _t {\tilde{v}}_i - \partial _t v\Vert _{L^1(0,T;L^\Phi (\Omega ))} +\lim _{i \rightarrow \infty } \Vert \partial _t e^{-\frac{t}{h_i}}(v(0)-v_{o,i})\Vert _{L^1(0,T;L^\Phi (\Omega ))} \\&\quad = \lim _{i \rightarrow \infty } \Vert \partial _t {\tilde{v}}_i - \partial _t v\Vert _{L^1(0,T;L^\Phi (\Omega ))} +\lim _{i \rightarrow \infty } \int _0^T \tfrac{1}{h_i}e^{-\frac{t}{h_i}} \,\mathrm {d}t\cdot \Vert v(0)-v_{o,i}\Vert _{L^\Phi (\Omega )} \\&\quad \le \lim _{i \rightarrow \infty } \Vert \partial _t {\tilde{v}}_i - \partial _t v\Vert _{L^1(0,T;L^\Phi (\Omega ))} +\lim _{i \rightarrow \infty }\Vert v(0)-v_{o,i}\Vert _{L^\Phi (\Omega )} =0. \end{aligned}$$

Finally, we establish the assertion

$$\begin{aligned} Dv_i \rightarrow Dv \text{ in } L^p(\Omega _T,\mathbb {R}^n) \text{ as } i \rightarrow \infty . \end{aligned}$$
(5.5)

To this end, we consider the mollification according to (2.7) with zero initial values, i.e., \(v_i^o := v_i -e^{-\frac{t}{h_i}} v_{o,i}\). Lemma 2.15 implies

$$\begin{aligned} Dv_i^o \rightarrow Dv \text{ in } L^p(\Omega _T,\mathbb {R}^n) \text{ as } i \rightarrow \infty . \end{aligned}$$
(5.6)

Further, observe that

$$\begin{aligned} \iint _{\Omega _T} \left| D\big (e^\frac{t}{h_i} v_{o,i}\big )\right| ^p \,\mathrm {d}x\mathrm {d}t= \int _0^T e^{-\frac{pt}{h_i}} \,\mathrm {d}t\int _\Omega |Dv_{o,i}|^p \,\mathrm {d}x\le \tfrac{h_i}{p} \int _\Omega |Dv_{o,i}|^p \,\mathrm {d}x. \end{aligned}$$

Using Young’s inequality for convolutions, we estimate the last integral. This yields

$$\begin{aligned} \int _\Omega |Dv_{o,i}|^p \,\mathrm {d}x&= \int _\Omega |(\chi _{\Omega \setminus \Omega _{\varepsilon _i}} g_o + \chi _{\Omega _{\varepsilon _i}} v(0)) *D\eta _{\varepsilon _i}|^p \,\mathrm {d}x\\&\le \Vert \chi _{\Omega \setminus \Omega _{\varepsilon _i}} g_o + \chi _{\Omega _{\varepsilon _i}} v(0)\Vert _{L^1(\Omega )}^p \Vert D\eta _{\varepsilon _i}\Vert _{L^p(\mathbb {R}^n)}^p \\&\le (\Vert 1\Vert '_{L^{\Phi ^*}(\Omega )})^p \big [\Vert g_o\Vert _{L^\Phi (\Omega )} + \Vert v(0)\Vert _{L^\Phi (\Omega )}\big ]^p \varepsilon _i^{-(n+1)p+n} \int _{\mathbb {R}^n} |D\eta |^p \,\mathrm {d}x\\&\le c \big [\Vert g_o\Vert _{L^\Phi (\Omega )} + \Vert v(0)\Vert _{L^\Phi (\Omega )}\big ]^p \varepsilon _i^{-(n+1)p} \end{aligned}$$

with \(c=c(n,p,m,|\Omega |)\). Joining the two preceding estimates and recalling the definition of \(\varepsilon _i\), we obtain that

$$\begin{aligned} \iint _{\Omega _T}&\left| D\big (e^\frac{t}{h_i} v_{o,i}\big )\right| ^p \,\mathrm {d}x\mathrm {d}t\\&\le c \big [\Vert g_o\Vert _{L^\Phi (\Omega )} + \Vert v(0)\Vert _{L^\Phi (\Omega )}\big ]^p \varepsilon _i^{-(n+1)p} h_i \\&= c \big [\Vert g_o\Vert _{L^\Phi (\Omega )} + \Vert v(0)\Vert _{L^\Phi (\Omega )}\big ]^p \sqrt{h_i} \rightarrow 0 \end{aligned}$$

in the limit \(i \rightarrow \infty\). Finally, combining the preceding assertion with (5.6), we obtain (5.5).

Solutions corresponding to the approximations

As we have shown in the previous section, the approximations \(g_i\) and \(\psi _i\) satisfy the assumptions of Theorem 4.2. Consequently, for each \(i \in \mathbb {N}\) there exists a variational solution \(u_i \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g_i + L^p(0,T;W^{1,p}_0(\Omega ))\big )\) corresponding to \(g_i\) and \(\psi _i\). By Lemma 2.20 with \(v=g_i\) we deduce the energy bound

$$\begin{aligned} \tfrac{1}{2} \sup _{t \in [0,T]} \varrho _\Omega (u_i(t)) + \iint _{\Omega _T} f(Du_i) \,\mathrm {d}x\le C, \end{aligned}$$
(5.7)

where the constant C is given by

$$\begin{aligned} C&:= \tfrac{2^\ell }{m} + \tfrac{1}{2} \sup _{i \in \mathbb {N}} \sup _{t \in [0,T]} \varrho _\Omega (g_i(t)) + c(\ell ,m) \sup _{i \in \mathbb {N}} \bigg (\int _0^T \Vert \partial _t g_i\Vert _{L^\Phi (\Omega )} \,\mathrm {d}t\bigg )^\frac{\ell +1}{\ell } \\&\quad + \tfrac{3}{\ell } \sup _{i \in \mathbb {N}} \iint _{\Omega _T} f(Dg_i) \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$

By (5.2) together with (1.6), (5.4) and (5.5) together with the growth condition (1.3), C is finite. Hence, there exists a limit map \(u \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}_0(\Omega )) \big)\) and a (not relabeled) subsequence such that

$$\begin{aligned} \left\{ \begin{array}{l} u_i \mathop {\rightharpoondown }\limits ^{*}u \text{ weakly }^* \text{ in } L^\infty (0,T;L^\Phi (\Omega )), \\ \hbox {} u_i \rightharpoondown u \hbox { weakly in } L^p(0,T;W^{1,p}(\Omega )) \end{array} \right. \end{aligned}$$
(5.8)

in the limit \(i \rightarrow \infty\). The obstacle condition is preserved as \(i \rightarrow \infty\). Indeed, by (5.8) and (5.2) applied to \(v_i=\psi _i\), we find that

$$\begin{aligned} \iint _{\Omega _T} (u-\psi )\varphi \,\mathrm {d}x\mathrm {d}t= \lim _{i \rightarrow \infty } \iint _{\Omega _T} (u_i-\psi _i)\varphi \,\mathrm {d}x\mathrm {d}t\ge 0 \quad \forall \varphi \in C^\infty _0(\Omega _T,\mathbb {R}_{\ge 0}). \end{aligned}$$

Convergence of solutions

In this step, we wish to establish

$$\begin{aligned} b(u_i) \mathop {\rightharpoondown }\limits ^{*}b(u) \text{ weakly }^* \text{ in } L^\infty (0,T;L^{\Phi ^*}(\Omega )) \text{ as } i \rightarrow \infty . \end{aligned}$$
(5.9)

Since \((b(u_i))_{i \in \mathbb {N}}\) is bounded in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\) by Lemma 2.3, (1.7) and (5.7), we know that there exists \(w \in L^\infty (0,T;L^{\Phi ^*}(\Omega ))\) such that (for a subsequence)

$$\begin{aligned} b(u_i) \mathop {\rightharpoondown }\limits ^{*}w \text{ weakly }^* \text{ in } L^\infty (0,T;L^{\Phi ^*}(\Omega )) \text{ as } i \rightarrow \infty . \end{aligned}$$
(5.10)

However, it remains to prove that w has the structure b(u). To this end, let \(\lambda >0\) and consider mollifications \([u_i-\psi _i]_\lambda\) and \([u-\psi ]_\lambda\) according to (2.7) with zero initial datum. Then, we define

$$\begin{aligned} w_{i,\lambda } := [u_i-\psi _i]_\lambda + \psi _i \text{ and } w_\lambda := [u-\psi ]_\lambda + \psi . \end{aligned}$$

By Lemma 2.15 we obtain that \(w_{i,\lambda } \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g_i + L^p(0,T;W^{1,p}(\Omega )) \big)\) and \(w_\lambda \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}(\Omega )) \big)\). Furthermore, (2.6) implies that

$$\begin{aligned} \partial _t [u_i-\psi _i]_\lambda = -\tfrac{1}{h_i}([u_i-\psi _i]_\lambda - (u_i-\psi _i)) \in L^\infty (0,T;L^\Phi (\Omega )) \cap L^p(\Omega _T). \end{aligned}$$

Since \(\psi _i \rightarrow \psi\) in \(L^p(\Omega _T)\), by (5.8)\(_2\) and Lemma 2.17, the sequence \(([u_i-\psi _i]_\lambda )_{i \in \mathbb {N}}\) is bounded in \(L^p(\Omega _T)\) for any fixed \(\lambda >0\). Further, by (5.5), (5.8)\(_2\) and Lemma 2.17

$$\begin{aligned} w_{i,\lambda } \rightharpoondown w_\lambda \text{ weakly } \text{ in } L^p(0,T;W^{1,p}(\Omega )) \text{ as } i \rightarrow \infty \end{aligned}$$
(5.11)

holds true for fixed \(\lambda >0\). Therefore, we conclude from Rellich’s theorem that

$$\begin{aligned} w_{i,\lambda } \rightarrow w_\lambda \text{ in } L^p(\Omega _T) \text{ as } i \rightarrow \infty \end{aligned}$$
(5.12)

for any fixed \(\lambda >0\). Here, we did not have to pass to a subsequence, since the limit is determined by (5.11). Since \(u_i \ge \psi _i\) for any \(i \in \mathbb {N}\), we know that \(w_{i,\lambda } \ge \psi _i\). Using \(w_{i,\lambda }\) as comparison map in the variational inequality for \(u_i\) leads us to

$$\begin{aligned}&-\iint _{\Omega _T} \partial _t w_{i,\lambda } \big (b(w_{i,\lambda }) - b(u_i)\big ) \,\mathrm {d}x\mathrm {d}t\\&\quad = \tfrac{1}{\lambda } \iint _{\Omega _T} (w_{i,\lambda }-u_i)\big (b(w_{i,\lambda }) - b(u_i)\big ) \,\mathrm {d}x\mathrm {d}t-\iint _{\Omega _T} \partial _t \psi _i \big (b(w_{i,\lambda }) - b(u_i)\big ) \,\mathrm {d}x\mathrm {d}t\\&\quad \le \iint _{\Omega _T} f(Dw_{i,\lambda }) \,\mathrm {d}x\mathrm {d}t+{\mathfrak {B}}[g_{o,i},\psi _i(0)]. \end{aligned}$$

The preceding inequality is equivalent to

$$\begin{aligned} \iint _{\Omega _T}&(w_{i,\lambda }-u_i)\big (b(w_{i,\lambda }) - b(u_i)\big ) \,\mathrm {d}x\mathrm {d}t\nonumber \\&\le \lambda \iint _{\Omega _T} \partial _t \psi _i \big (b(w_{i,\lambda })- b(u_i)\big ) \,\mathrm {d}x\mathrm {d}t+\lambda \iint _{\Omega _T} f(Dw_{i,\lambda }) \,\mathrm {d}x\mathrm {d}t\nonumber \\&\quad +\lambda {\mathfrak {B}}[g_{o,i},\psi _i(0)] \nonumber \\&=: \lambda \mathrm {I}_\lambda + \lambda \mathrm {II}_\lambda + \lambda \mathrm {III}, \end{aligned}$$
(5.13)

where the definition of \(\mathrm {I}_\lambda\), \(\mathrm {II}_\lambda\) and \(\mathrm {III}\) is clear in this context. By the generalized Hölder's inequality (1.8), (2.5) and Lemma 2.3, we estimate

$$\begin{aligned} |\mathrm {I}_\lambda |&\le \Vert \partial _t \psi _i\Vert _{L^1(0,T;L^\Phi (\Omega ))} \big [\sup _{t \in [0,T]} \Vert b(w_{i,\lambda })\Vert '_{L^{\Phi ^*}(\Omega )} +\sup _{t \in [0,T]} \Vert b(u_i)\Vert '_{L^{\Phi ^*}(\Omega )} \big ] \\&\le \Vert \partial _t \psi _i\Vert _{L^1(0,T;L^\Phi (\Omega ))} \big [2+ \sup _{t \in [0,T]} \varrho _\Omega ^*(b(w_{i,\lambda }))^\frac{1}{\ell +1} +\sup _{t \in [0,T]} \varrho _\Omega ^*(b(u_i))^\frac{1}{\ell +1} \big ] \\&\le \Vert \partial _t \psi _i\Vert _{L^1(0,T;L^\Phi (\Omega ))} \big [2+ \sup _{t \in [0,T]} (m \varrho _\Omega (w_{i,\lambda }))^\frac{1}{\ell +1} +\sup _{t \in [0,T]} (m \varrho _\Omega (u_i))^\frac{1}{\ell +1} \big ]. \end{aligned}$$

By (5.4) applied to \(v_i=\psi _i\), the first factor on the right-hand side of the preceding inequality stays bounded in the limit \(i \rightarrow \infty\). Further, by the energy bound (5.7) we know that \(\sup _{t \in [0,T]}\int _\Omega \Phi (u_i) \,\mathrm {d}x\le 2C\) for any \(i \in \mathbb {N}\). By Lemma 2.15, (5.2) and (5.8)\(_1\) we find that

$$\begin{aligned} \sup _{i \in \mathbb {N}}\Vert w_{i,\lambda }\Vert _{L^\infty (0,T;L^\Phi (\Omega ))}&\le \sup _{i \in \mathbb {N}} \Vert [u_i-\psi _i]_\lambda \Vert _{L^\infty (0,T;L^\Phi (\Omega ))} +\sup _{i \in \mathbb {N}}\Vert u_i\Vert _{L^\infty (0,T;L^\Phi (\Omega ))} \\&\le \sup _{i \in \mathbb {N}}\Vert u_i-\psi _i\Vert _{L^\infty (0,T;L^\Phi (\Omega ))} +\sup _{i \in \mathbb {N}}\Vert u_i\Vert _{L^\infty (0,T;L^\Phi (\Omega ))} <\infty . \end{aligned}$$

Consequently, by (1.7), we conclude that \(\sup _{t \in [0,T]}\int _\Omega \Phi (w_{i,\lambda }) \,\mathrm {d}x\) is bounded by a constant independent of \(i \in \mathbb {N}\) and \(\lambda >0\). Altogether, we obtain that

$$\begin{aligned} |\mathrm {I}_\lambda | \le c \end{aligned}$$
(5.14)

with a constant c independent of i and \(\lambda\). Next, by the growth condition (1.3), Lemma 2.15 and (5.5), we deduce that

$$\begin{aligned} |\mathrm {II}_\lambda |&\le L\int _\Omega \big [1+|Dw_{i,\lambda }|^p\big ] \,\mathrm {d}x\mathrm {d}t\nonumber \\&\le L|\Omega |T +2^{p-1}L\big [\Vert [Du_i-D\psi _i]_\lambda \Vert _{L^p(\Omega _T,\mathbb {R}^n)}^p +\Vert D\psi _i\Vert _{L^p(\Omega _T,\mathbb {R}^n)}^p\big ] \nonumber \\&\le L|\Omega |T +c(p)L\big [\Vert Du_i\Vert _{L^p(\Omega _T,\mathbb {R}^n)}^p +\Vert D\psi _i\Vert _{L^p(\Omega _T,\mathbb {R}^n)}^p\big ] \nonumber \\&\le L|\Omega |T +c(p)L\big [C + \sup _{i \in \mathbb {N}}\Vert D\psi _i\Vert _{L^p(\Omega _T,\mathbb {R}^n)}^p\big ] <\infty . \end{aligned}$$
(5.15)

Finally, by (5.1), we have that \({\mathfrak {b}}[g_{o,i},\psi _i(0)] \rightarrow {\mathfrak {b}}[g_o,\psi (0)]\) a.e. in \(\Omega _T\). Further, by Lemma 2.3, the nonnegativity of \(b(g_{o,i})\) and \(\psi _i(0)\) and (5.1), we find the dominating function

$$\begin{aligned} 0&\le {\mathfrak {b}}[g_{o,i},\psi _i(0)] \le \Phi (\psi _i(0)) + \Phi ^*(b(g_{o,i})) \le \Phi (\psi _i(0)) + m\Phi (g_{o,i}) \\&\rightarrow \Phi (\psi (0)) + m\Phi (g_o). \end{aligned}$$

Consequently, by a version of the dominated convergence theorem (cf. [16, Theorem 1.20]), we know that \({\mathfrak {B}}[g_{o,i},\psi _i(0)] \rightarrow {\mathfrak {B}}[g_o,\psi (0)]\) as \(i \rightarrow \infty\) and hence

$$\begin{aligned} \sup _{i \in \mathbb {N}} \mathrm {III} < \infty . \end{aligned}$$
(5.16)

Inserting (5.14), (5.15) and (5.16) into (5.13), we obtain

$$\begin{aligned} \iint _{\Omega _T} (w_{i,\lambda }-u_i)\big (b(w_{i,\lambda }) - b(u_i)\big ) \,\mathrm {d}x\mathrm {d}t\le c\lambda \end{aligned}$$

with a constant c independent of \(i \in \mathbb {N}\) and \(\lambda >0\). In order to bound the left-hand side of the preceding inequality from below, we apply Lemma 2.9, Hölder’s inequality and Lemma 2.7. This yields

$$\begin{aligned}&\varrho _{\Omega _T}(w_{i,\lambda } - u_i) \le \iint _{\Omega _T} |\Phi (w_{i,\lambda }) - \Phi (u_i)| \,\mathrm {d}x\mathrm {d}t\\&\quad \le \bigg (\iint _{\Omega _T} \big |\sqrt{\Phi (w_{i,\lambda })} + \sqrt{\Phi (u_i)}\big |^2 \,\mathrm {d}x\mathrm {d}t\bigg )^\frac{1}{2} \bigg (\iint _{\Omega _T} \big |\sqrt{\Phi (w_{i,\lambda })} - \sqrt{\Phi (u_i)}\big |^2 \,\mathrm {d}x\mathrm {d}t\bigg )^\frac{1}{2} \\&\quad \le c \big [ \varrho _{\Omega _T}(w_{i,\lambda })^\frac{1}{2} + \varrho _{\Omega _T}(u_i)^\frac{1}{2} \big ] \bigg (\iint _{\Omega _T} (w_{i,\lambda }-u_i)\big (b(w_{i,\lambda }) - b(u_i)\big ) \,\mathrm {d}x\mathrm {d}t\bigg )^\frac{1}{2} \end{aligned}$$

We have already established that the first factor on the right-hand side of the preceding inequality is bounded independent of i and \(\lambda\). Therefore, we obtain that

$$\begin{aligned} \varrho _{\Omega _T}(w_{i,\lambda } - u_i) \le c\lambda ^\frac{1}{2}, \end{aligned}$$

where the constant is independent of i and \(\lambda\). Thus, by (1.6) and Lemma 2.8 for any \(\delta >0\), there exists \(\lambda _o>0\) such that

$$\begin{aligned} \Vert w_{i,\lambda } - u_i\Vert _{L^{\ell +1}(\Omega _T)} < \tfrac{\delta }{2} \end{aligned}$$
(5.17)

holds true for any \(0<\lambda <\lambda _o\). Decreasing \(\lambda _o\) if necessary, by Lemma 2.15, we may assume that

$$\begin{aligned} \Vert w_\lambda - u\Vert _{L^{\ell +1}(\Omega _T)} < \tfrac{\delta }{2} \end{aligned}$$
(5.18)

holds true for any \(0<\lambda <\lambda _o\) as well. Finally, we abbreviate \(q := \min \{p,\ell +1\}\). Combining (5.12), (5.17) and (5.18) with the choice \(0<\lambda <\lambda _o\), we infer

$$\begin{aligned}&\limsup _{i \rightarrow \infty } \Vert u_i-u\Vert _{L^q(\Omega _T)} \\&\quad \le \limsup _{i \rightarrow \infty } \big [\Vert u_i-w_{i,\lambda }\Vert _{L^q(\Omega _T)} +\Vert w_{i,\lambda }-w_\lambda \Vert _{L^q(\Omega _T)} +\Vert w_\lambda -u\Vert _{L^q(\Omega _T)}\big ] \\&< \delta + \lim _{i \rightarrow \infty } \Vert w_{i,\lambda }-w_\lambda \Vert _{L^q(\Omega _T)} = \delta . \end{aligned}$$

Since \(\delta >0\) was arbitrary, the preceding consideration yields

$$\begin{aligned} u_i \rightarrow u \text{ in } L^{\min \{p,\ell +1\}}(\Omega _T) \text{ as } i \rightarrow \infty . \end{aligned}$$

Passing to a subsequence, we further have that \(u_i \rightarrow u\) a.e. in \(\Omega _T\) as \(i \rightarrow \infty\). Together with (5.10), this proves (5.9).

Conclusion of the proof

Consider a comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). We define mollifications \(v_i\) to v as in Sect. 5.1. Since we are not able to pass to the limit \(i \rightarrow \infty\) in the variational inequality satisfied by \(u_i\) and \(v_i\) directly, we integrate over \(\tau \in (t_o,t_o+\delta )\) for some \(\delta \in (0,T)\) and \(t_o \in (0,T-\delta ]\). This leads to

$$\begin{aligned} \iint _{\Omega _{t_o}}&f(Du_i) \,\mathrm {d}x\mathrm {d}t+\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } {\mathfrak {B}}[u_i(\tau ),v_i(\tau )] \,\mathrm {d}\tau \nonumber \\&\le \iint _{\Omega _{t_o+\delta }} f(Dv_i) \,\mathrm {d}x\mathrm {d}t+\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \iint _{\Omega _\tau } \partial _t v_i \big (b(v_i)-b(u_i)\big ) \,\mathrm {d}x\mathrm {d}t\mathrm {d}\tau \nonumber \\&\quad +{\mathfrak {B}}[g_{o,i},v_i(0)]. \end{aligned}$$
(5.19)

In the following, we consider the terms of (5.19) separately. Since f is convex and satisfies the growth condition (1.3), we conclude from (5.8)\(_2\) that

$$\begin{aligned} \iint _{\Omega _{t_o}} f(Du) \,\mathrm {d}x\mathrm {d}t\le \liminf _{i \rightarrow \infty } \iint _{\Omega _{t_o}} f(Du_i) \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$
(5.20)

In order to treat the second term on the left-hand side of (5.19), note that

$$\begin{aligned} {\mathfrak {b}}[u_i(\tau ),v_i(\tau )] = \Phi (v_i(\tau )) + \Phi ^*(b(u_i(\tau ))) - b(u_i(\tau ))v_i(\tau ). \end{aligned}$$

Since \(\Phi\) is convex and the functional \(\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \int _\Omega \Phi (\cdot ) \,\mathrm {d}x\mathrm {d}\tau\) is continuous with respect to strong convergence in \(L^\Phi (\Omega _T)\) by (1.6), we conclude that it is lower semicontinuous with respect to weak convergence in \(L^\Phi (\Omega _T)\), cf. [14, Corollary 3.9]. Similarly, we find that \(\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \int _\Omega \Phi ^*(\cdot ) \,\mathrm {d}x\mathrm {d}\tau\) is lower semicontinuous with respect to weak convergence in \(L^{\Phi ^*}(\Omega _T)\). Thus, as a consequence of (5.2) and (5.9), we obtain that

$$\begin{aligned} \mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } {\mathfrak {B}}[u(\tau ),v(\tau )] \,\mathrm {d}\tau \le \liminf _{i \rightarrow \infty } \mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } {\mathfrak {B}}[u_i(\tau ),v_i(\tau )] \,\mathrm {d}\tau . \end{aligned}$$
(5.21)

By the Lipschitz condition (1.4) and (5.5), we have that

$$\begin{aligned} \lim _{i \rightarrow \infty } \iint _{\Omega _{t_o+\delta }} f(Dv_i) \,\mathrm {d}x\mathrm {d}t= \iint _{\Omega _{t_o+\delta }} f(Dv) \,\mathrm {d}x\mathrm {d}t. \end{aligned}$$
(5.22)

Next, by (5.3), (5.4) and (5.8)\(_1\), we conclude that

$$\begin{aligned}&\lim _{i \rightarrow \infty } \mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \iint _{\Omega _\tau } \partial _t v_i \big (b(v_i)-b(u_i)\big ) \,\mathrm {d}x\mathrm {d}t\mathrm {d}\tau \nonumber \\&\quad = \mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \iint _{\Omega _\tau } \partial _t v \big (b(v)-b(u)\big ) \,\mathrm {d}x\mathrm {d}t\mathrm {d}\tau . \end{aligned}$$
(5.23)

Finally, by (5.1) applied to \(g_{o,i}\) and \(v_i(0)\) and the dominated convergence theorem, we infer

$$\begin{aligned} \lim _{i \rightarrow \infty }{\mathfrak {B}}[g_{o,i},v_i(0)] = {\mathfrak {B}}[g_o,v(0)]. \end{aligned}$$
(5.24)

Collecting (5.20), (5.21), (5.22), (5.23) and (5.24), we deduce from (5.19) that

$$\begin{aligned} \iint _{\Omega _{t_o}}&f(Du) \,\mathrm {d}x\mathrm {d}t+\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } {\mathfrak {B}}[u(\tau ),v(\tau )] \,\mathrm {d}\tau \nonumber \\&\le \iint _{\Omega _{t_o+\delta }} f(Dv) \,\mathrm {d}x\mathrm {d}t+\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \iint _{\Omega _\tau } \partial _t v \big (b(v)-b(u)\big ) \,\mathrm {d}x\mathrm {d}t\mathrm {d}\tau +{\mathfrak {B}}[g_o,v(0)] \end{aligned}$$

holds true for any admissible comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). Passing to the limit \(\delta \downarrow 0\), we conclude that u is a variational solution to (1.2) in the sense of Definition 1.1. \(\square\)

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L. Schätzler has been supported by Studienstiftung des deutschen Volkes.

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Schätzler, L. The obstacle problem for degenerate doubly nonlinear equations of porous medium type. Annali di Matematica 200, 641–683 (2021). https://doi.org/10.1007/s10231-020-01008-y

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Keywords

  • Porous medium equation
  • Doubly nonlinear equations
  • Existence
  • Variational solutions
  • Minimizing movements

Mathematics Subject Classification

  • 35K86
  • 49J40
  • 49J45