Study of weak solutions for degenerate parabolic inequalities with nonstandard conditions

Abstract

In this paper, we study the degenerate parabolic variational inequalities in a bounded domain. By solving a series of penalty problems, the existence and uniqueness of the solutions in the weak sense are proved by the energy method and a limit process.

1 Introduction

Let $0 < T < \infty$ and $\Omega \subset {{\mathrm{{R}}}_{N}} (N \ge 2)$ be a bounded simple domain with appropriately smooth boundary ∂Ω. In this article, we consider the following quasilinear degenerate parabolic inequalities:

$$\begin{aligned} \textstyle\begin{cases} \min \{ Lu,u(x,0) - {u_{0}}\} = 0, & (x,t) \in {Q_{T}}, \\ u(x,t) = 0, & (x,t) \in {\Gamma _{T}}, \\ u(x,0) = {u_{0}}, & x \in \Omega, \end{cases}\displaystyle \end{aligned}$$

(1.1)

with

$$\begin{aligned} Lu = {u_{t}} - u{\mathrm{{div}}} \bigl(a(u){ \vert {\nabla u} \vert ^{p(x,t) - 2}} \nabla u \bigr) - \gamma { \vert {\nabla u} \vert ^{p(x,t)}} - f(x,t), \end{aligned}$$

(1.2)

where ${Q_{T}} = \Omega \times (0,T]$, ${\mathrm{{a}}}(u) = {u^{\sigma }} + {d_{0}}$, and ${\Gamma _{T}}$ is the lateral boundary of cylinder $Q_{T}$.

In applications, Problem (1.1) arises in the model of American option pricing in the Black–Scholes models. We refer to [1–4] for the financial background of parabolic inequalities. Among them, the most interesting research topic is to construct different types of variational parabolic inequalities and analyze the existence and uniqueness for their solutions (see, for example, [3–10] and the references therein). In 2014, the authors in [5] discussed the problem

$$\begin{aligned} \textstyle\begin{cases} {u_{t}} - Lu - F(u,x,t) \ge 0 &\text{in } {Q_{T}}, \\ u(x,t) \ge {u_{0}}(x) &\text{in } \Omega, \\ ( {{u_{t}} - Lu - F(u,x,t)} ) \cdot ( {u(x,t) - {u_{0}}(x)} ) = 0 &\text{in } {Q_{T}}, \\ u(x,0) = {u_{0}}(x) &\text{in } \Omega, \\ u(x,t) = 0 &\text{on } {\Gamma _{T}}, \end{cases}\displaystyle \end{aligned}$$

with second-order elliptic operator

$$\begin{aligned} Lu = - u{\mathrm{{div}}} \bigl(a(u){ \vert {\nabla u} \vert ^{p(x,t) - 2}} \nabla u \bigr) - \gamma { \vert {\nabla u} \vert ^{p(x,t)}} - f(x,t). \end{aligned}$$

They proved the existence and uniqueness of a solution to this problem with some restrictions on $u_{0}$, F, and L. Later, the authors in [6, 7] extended the relative conclusions with the assumption that $a(u)$ is a constant, and $p(x) =2$. The authors discussed the existence and numerical algorithm of the solution.

To the best of our knowledge, the existence and uniqueness of this problem with the assumption that $p(x,t)$ are variables have been less studied. We cannot easily apply the method in [6, 7] to the case that $p(x,t)$ and $a(u)$ are not constants.

The aim of this paper is to study the existence and uniqueness of solutions for a degenerate parabolic variational inequality problem. Throughout the paper, we assume that the exponent $p(x, t)$ is continuous in $Q_{T}$ with a logarithmic module of a country:

$$\begin{aligned} 1 < {p^{-} } < p(x,t) < {p^{+} } < \infty, \end{aligned}$$

(1.3)

where ${p^{-} } = \mathop {\inf } _{(x,t) \in {Q_{T}}} p(x,t)$ and ${p^{+} } = \mathop {\sup } _{(x,t) \in {Q_{T}}} p(x,t)$.

The outline of this paper is as follows: In Sect. 2, we introduce the function spaces of Orlicz–Sobolev type, give the definition of the weak solution to the problem, and state our main theorems. In Sect. 3, we give some estimates of the penalty problem (approximating problem). Section 4 proves the existence and uniqueness of the solution obtained in Sect. 2.

2 The main results of weak solutions

In this section, we recall some useful definitions and known results, which can be found in [11–14]. Set

$$\begin{aligned} &{L^{p(x,t)}}({Q_{T}}) = \biggl\{ u(x,t)|u \text{ is measurable in } {Q_{T}},{A_{p(\cdot)}}(u) = \int { \int _{{Q_{T}}} {{{ \vert u \vert }^{p(x,t)}} {\, \mathrm{d}}x{\,\mathrm{d}}t} } \biggr\} ,\\ &{ \Vert u \Vert _{p(\cdot)}} = \inf \bigl\{ \lambda > 0,{A_{p(\cdot)}}(u/ \lambda ) \le 1 \bigr\} ,\\ &{V_{t}}(\Omega ) = \bigl\{ u| {u \in {L^{2}}(\Omega ) \cap W_{0}^{1,1}, \vert {\nabla u} \vert \in {L^{p(x,t)}}( \Omega )} \bigr\} ,\\ &{ \Vert u \Vert _{{V_{t}}(\Omega )}} = { \Vert u \Vert _{2, \Omega }} + { \Vert {\nabla u} \Vert _{p( \cdot,t),\Omega }},\\ &W({Q_{T}}) = \bigl\{ u:[0,T] \to {V_{t}}( \Omega )| {u \in {L^{2}}({Q_{T}}) \cap W_{0}^{1,1},} \\ &\vert {\nabla u} \vert \in {L^{p(x,t)}}({Q_{T}}),u = 0 \text{ on } {\Gamma _{T}}\bigr\} , \\ &{ \Vert u \Vert _{{W_{t}}({Q_{T}})}} = { \Vert u \Vert _{2,{Q_{T}}}} + { \Vert {\nabla u} \Vert _{p( \cdot,t),{Q_{T}}}} \end{aligned}$$

and denote by $W'({Q_{T}})$ the dual of $W({Q_{T}})$ with respect to the inner product in ${L^{2}}({Q_{T}})$.

In the spirit of [3] and [4], we introduce the following maximal monotone graph

$$\begin{aligned} G(x) = \textstyle\begin{cases} 0,& x > 0, \\ \theta,& x =0, \end{cases}\displaystyle \end{aligned}$$

(2.1)

where $\theta \in [0,M)$ and M depends only on $|u_{0}|_{\infty}$.

The purpose of the paper is to obtain the existence and uniqueness of weak solutions of (1.1). Let $B = W({Q_{T}}) \cap {L^{\infty }}(0,T;{L^{\infty }}(\Omega ))$, and the weak solution is defined as:

Definition 2.1

A pair is called a weak solution of problem (1.1), if (a) $u(x,t) \ge {u_{0}}(x)$, (b) $u(x,0) = {u_{0}}(x)$, (c) $\xi \in G(u - {u_{0}})$, (d) for every test function $\phi \in Z \equiv \{ \eta (z):\eta \in W({Q_{T}}) \cap {L^{\infty }}(0,T;{L^{2}}( \Omega )),{\eta _{t}} \in W({Q_{T}})\}$ and every ${t_{1}},{t_{2}} \in [0,T]$ the following identity holds:

$$\begin{aligned} \begin{aligned} &\int _{{t_{1}}}^{{t_{2}}} { \int _{\Omega }{u \cdot {\phi _{t}} - a(u){{ \vert { \nabla u} \vert }^{p(x,t) - 2}}\nabla u\nabla \phi - \bigl(a(u) - \gamma \bigr){{ \vert {\nabla u} \vert }^{p(x,t)}}\phi {\,\mathrm{d}}x{\, \mathrm{d}}t} } \\ &\quad{}+ \int _{{t_{1}}}^{{t_{2}}} { \int _{\Omega }{ f(x,t)\phi + \xi \phi { \,\mathrm{d}}x{\, \mathrm{d}}t} } = \int _{\Omega }{u\phi {\,\mathrm{d}}x} \int _{{t_{1}}}^{{t_{2}}}. \end{aligned} \end{aligned}$$

(2.2)

Our main results are the following two theorems.

Theorem 2.1

Let us satisfy conditions (1.3). If the following conditions hold:

• (H1) $\max \{ 1,\frac{{2N}}{{N + 1}}\} < {p^{-} } < N$, $2 \le \sigma < \frac{{2{p^{+} }}}{{{p^{+} } - 1}}$, $0 < \gamma < {d_{0}}$, and

• (H2) ${u_{0}}(x) \ge 0, f \ge 0, { \Vert {{u_{0}}} \Vert _{ \infty,\Omega }} + \int _{0}^{T} {{{ \Vert {f(x,t)} \Vert }_{ \infty,\Omega }}{\,\mathrm{d}}t} + \vert \Omega \vert \cdot T = K(T) < \infty$,

then Problem (1.1) has at least one weak solution in the sense of Definition 2.1.

Theorem 2.2

Suppose that the conditions in Theorem 2.1are fulfilled and $p^{+} \geq 2$. Then, Problem (1.1) admits a unique solution in the sense of Definition 2.1.

3 Penalty problems

In this section, we consider a family of auxiliary parabolic problems

$$\begin{aligned} \textstyle\begin{cases} {L_{\varepsilon }}{u_{\varepsilon }} + \beta ({u_{\varepsilon }} - {u_{0}}) = 0,& (x,t) \in {Q_{T}}, \\ {u_{\varepsilon }}(x,t) = \varepsilon, & (x,t) \in { \Gamma _{T}}, \\ {u_{\varepsilon }}(x,0) = {u_{0}} + \varepsilon, & x \in \Omega, \end{cases}\displaystyle \end{aligned}$$

(3.1)

with

$$\begin{aligned} {L_{\varepsilon }} {u_{\varepsilon }} = {u_{\varepsilon }}_{t} - {u_{ \varepsilon }} \cdot {\mathrm{{div}}} \bigl(a({u_{\varepsilon }}){ \vert { \nabla {u_{\varepsilon }}} \vert ^{p(x) - 2}}\nabla {u_{ \varepsilon }} \bigr) - \gamma { \vert {\nabla {u_{\varepsilon }}} \vert ^{p(x,t)}} - f(x,t), \end{aligned}$$

(3.2)

${\beta _{\varepsilon }}( \cdot )$ is the penalty function satisfying

$$\begin{aligned} \begin{aligned} &\varepsilon \in (0,1),\qquad{\beta _{\varepsilon }}( \cdot ) \in {C^{2}}({ \mathrm{{R}}}),\qquad{\beta _{\varepsilon }}(x) \le 0,\qquad{\beta _{\varepsilon }}(0) = - 1, \\ &{{\beta '}_{\varepsilon }}(0) \ge 0,\qquad{{\beta ''}_{\varepsilon }}(0) \ge 0,\qquad\mathop {\lim } _{x \to 0 + } \beta (x) = \textstyle\begin{cases} 0,&x > - 0, \\ - 1,&x = 0. \end{cases}\displaystyle \end{aligned} \end{aligned}$$

(3.3)

With a similar method as in [8], we may prove that the regularized problem has a unique weak solution

$$\begin{aligned} {u_{\varepsilon }}(x,t) \in W({Q_{T}}) \cap {L^{2}}({Q_{T}}),\quad{ \partial _{t}} {u_{\varepsilon }}(x,t) \in W\prime ({Q_{T}}), \end{aligned}$$

satisfying the following integral identities

$$\begin{aligned} \begin{aligned} &\int _{{t_{1}}}^{{t_{2}}} { \int _{\Omega }{{u_{\varepsilon }} \cdot { \phi _{t}} - a({u_{\varepsilon }}){u_{\varepsilon }} {{ \vert { \nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{ \varepsilon }}\nabla \phi - \bigl(a({u_{\varepsilon }}) - \gamma \bigr){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t)}}\phi {\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\quad= \int _{{t_{1}}}^{{t_{2}}} { \int _{\Omega }{ \bigl( {{\beta _{ \varepsilon }}({u_{\varepsilon }} - {u_{0}}) - f(x,t)} \bigr)\phi { \,\mathrm{d}}x{\,\mathrm{d}}t} } + \int _{\Omega }{{u_{\varepsilon }}\phi {\,\mathrm{d}}x} \int _{{t_{1}}}^{{t_{2}}} {} \end{aligned} \end{aligned}$$

(3.4)

and

$$\begin{aligned} \begin{aligned} &\int _{{t_{1}}}^{{t_{2}}} { \int _{\Omega }{{u_{\varepsilon }}_{t} \cdot \phi + a({u_{\varepsilon }}){u_{\varepsilon }} {{ \vert { \nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{ \varepsilon }}\nabla \phi + \bigl(a({u_{\varepsilon }}) - \gamma \bigr){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t)}}\phi {\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\quad= \int _{{t_{1}}}^{{t_{2}}} { \int _{\Omega }{ \bigl( {f(x,t) - { \beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr)\phi { \,\mathrm{d}}x{\,\mathrm{d}}t} }. \end{aligned} \end{aligned}$$

(3.5)

We start with two preliminary results that will be used several times below.

Lemma 3.1

Let $M(s) = { \vert s \vert ^{p(x,t) - 2}}s$, then $\forall \xi,\eta \in {{\mathrm{{R}}}^{N}}$

$$\begin{aligned} \begin{aligned} & \bigl( {M(\xi ) - M(\eta )} \bigr) \cdot (\xi - \eta ) \\ &\quad\ge \textstyle\begin{cases} {2^{ - p(x,t)}}{ \vert {\xi - \eta } \vert ^{p(x,t)}},& 2 \le p(x,t) < \infty, \\ (p(x,t) - 1){ \vert {\xi - \eta } \vert ^{2}}{ ( {{{ \vert \xi \vert }^{p(x,t)}} + {{ \vert \eta \vert }^{p(x,t)}}} )^{ \frac{{p(x,t) - 2}}{{p(x,t)}}}},& 1 \le p(x,t) < 2. \end{cases}\displaystyle \end{aligned} \end{aligned}$$

Proof

The proof can be found in [15]. □

Lemma 3.2

(Comparison principle)

Assume $2 < \sigma < \frac{{2{p^{+} }}}{{{p^{+} } - 1}}$, ${p^{+} } \ge 2$, u and v are in $W({Q_{T}}) \cap {L^{\infty }}(0,T;{L^{ \infty }}(\Omega ))$. If ${L_{\varepsilon }}u \ge {L_{\varepsilon }}v$ in $Q_{T}$ and $u(x,t) \le v(x,t)$ on $\partial {Q_{T}}$, then $u(x,t) \le v(x,t)$ in $Q_{T}$.

Proof

We argue by contradiction. Suppose $u(x,t)$ and $v(x,t)$ satisfy ${L_{\varepsilon }}u \ge {L_{\varepsilon }}v$ in ${Q_{T}}$ and there is a $\delta > 0$ such that for $0 < \tau \le T$, $w = u - v$ on the set

$$\begin{aligned} {\Omega _{\delta }} = \Omega \cap \bigl\{ {x:w(x,t) > \delta } \bigr\} \end{aligned}$$

and $\mu ({\Omega _{\delta }}) > 0$. Let

$$\begin{aligned} {F_{\varepsilon }}(\xi ) = \textstyle\begin{cases} \frac{1}{{\alpha - 1}}{\varepsilon ^{1 - \alpha }} - \frac{1}{{\alpha - 1}}{\xi ^{1 - \alpha }} &\text{if }\xi > \varepsilon, \\ 0 &\text{if }\xi \le \varepsilon, \end{cases}\displaystyle \end{aligned}$$

(3.6)

where $\delta > {\mathrm{{2}}}\varepsilon > 0$ and $\alpha = \frac{\sigma }{2}$. Let a test function $\xi = {F_{\varepsilon }}(w) \in Z$ in (3.4),

$$\begin{aligned} \begin{aligned} 0 \ge{}& \int { \int _{{Q_{T}}} {{w_{t}} {F_{\varepsilon }}(w) + a(v)v \bigl( { \bigl\vert \nabla u{|^{p(x,t) - 2}}\nabla u - \bigr\vert \nabla v{|^{p(x,t) - 2}} \nabla v} \bigr)\nabla {F_{\varepsilon }}(w){\,\mathrm{d}}x{\, \mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u)u - a(v)v\bigr]|\nabla u{|^{p(x,t) - 2}}\nabla u\nabla {F_{\varepsilon }}(w){\,\mathrm{d}}x{\, \mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u) - \gamma \bigr] \bigl\vert \nabla u{|^{p(x,t)}} - \bigl[a(v) - \gamma \bigr] \bigr\vert \nabla v{|^{p(x,t)}}} } \\ ={}& {J_{1}} + {J_{2}} + {J_{3}} + {J_{4}}, \end{aligned} \end{aligned}$$

(3.7)

where ${Q_{T,\varepsilon }} = \{ (x,t) \in {Q_{T}}|w > \varepsilon \}$,

$$\begin{aligned} &{J_{1}} = \int { \int _{{Q_{T}}} {{w_{t}} {F_{\varepsilon }}(w){\, \mathrm{d}}x{ \,\mathrm{d}}t} },\\ &{J_{2}} = \int { \int _{{Q_{T}}} {a(v)v \bigl( { \bigl\vert \nabla u{|^{p(x,t) - 2}} \nabla u - \bigr\vert \nabla v{|^{p(x,t) - 2}}\nabla v} \bigr)\nabla w{\,\mathrm{d}}x{ \,\mathrm{d}}t} },\\ &{J_{3}} = \int { \int _{{Q_{T}}} { \bigl[a(u)u - a(v)v \bigr]|\nabla u{|^{p(x,t) - 2}} \nabla u\nabla {F_{\varepsilon }}(w){\,\mathrm{d}}x{\, \mathrm{d}}t} },\\ &{J_{4}} = \int { \int _{{Q_{T}}} { \bigl[a(u) - \gamma \bigr] \bigl\vert \nabla u{|^{p(x,t)}} - \bigl[a(v) - \gamma \bigr] \bigr\vert \nabla v{|^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} }. \end{aligned}$$

Now, let ${t_{0}} = \inf \{ t \in (0,\tau ]:w > \varepsilon \}$, then we estimate $J_{1}$ as follows

$$\begin{aligned} \begin{aligned} {J_{1}}& = \int { \int _{{Q_{T}}} {{w_{t}} {F_{\varepsilon }}(w){\, \mathrm{d}}x{ \,\mathrm{d}}t} } = \int _{\Omega }\biggl( \int _{0}^{{t_{0}}} {{w_{t}} {F_{ \varepsilon }}(w){\,\mathrm{d}}t} + \int _{0}^{{t_{0}}} {{w_{t}} {F_{ \varepsilon }}(w){\,\mathrm{d}}t}\biggr)\,\mathrm{d}x \\ &\ge \int _{\Omega }{ \int _{\varepsilon }^{w} {{F_{ \varepsilon }}({\mathrm{{s}}}){ \,\mathrm{d}}s} } {\,\mathrm{d}}x \ge \int _{{\Omega _{ \delta }}} { \int _{\varepsilon }^{w} {{F_{\varepsilon }}({\mathrm{{s}}}){ \,\mathrm{d}}s} } {\,\mathrm{d}}x. \end{aligned} \end{aligned}$$

(3.8)

Let us first consider the case ${p^{-} } \ge 2$. By virtue of the first inequality of Lemma 3.1, we obtain

$$\begin{aligned} \begin{aligned} {J_{2}} &= \int { \int _{{Q_{T}}} {a(v)v\bigl({{ \vert {\nabla u} \vert }^{p(x,t) - 2}}\nabla u - {{ \vert {\nabla v} \vert }^{p(x,t) - 2}}\nabla v\bigr) \nabla w{\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\ge \int { \int _{{Q_{T}}} {a(v)v \cdot {w^{ - \alpha }} {2^{ - p(x,t)}} {{ \vert {\nabla w} \vert }^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &= {2^{ - {p^{+} }}} \int { \int _{{Q_{T}}} {a(v)v \cdot {w^{ - \alpha }} {{ \vert {\nabla w} \vert }^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} } > 0. \end{aligned} \end{aligned}$$

(3.9)

Noting that $\frac{{p(x,t)}}{{p(x,t) - 1}} \ge \frac{{p + }}{{p + - 1}} \ge { \sigma ^{2}} = \alpha > 1$ and applying Young’s inequality, we may estimate the integrand of $J_{3}$ in the following way

$$\begin{aligned} &\bigl\vert {\bigl[a(u)u - a(v)v \bigr]{w^{ - \alpha }} {{ \vert {\nabla w} \vert }^{p(x,t) - 2}}\nabla u \nabla w} \bigr\vert \\ \begin{aligned} &\quad= \biggl\vert { \biggl[ {(\delta + 1)w \int _{0}^{1} {{{\bigl(\theta u + (1 - \theta )v \bigr)}^{\sigma }}\,\mathrm{d}\theta } + {d_{0}}(u - v)} \biggr]{w^{ - \alpha }} {{ \vert {\nabla u} \vert }^{p(x,t) - 2}}\nabla u \nabla w} \biggr\vert \\ &\quad\le \frac{C}{{{w^{\alpha }}}} \biggl[ {\frac{{a(v)v}}{C}{{ \vert { \nabla w} \vert }^{p(x,t)}} + {C_{1}}\bigl(\delta,{d_{0}},K,{p^{\pm }} \bigr){{ \vert w \vert }^{p'(x,t)}} {{ \vert {\nabla u} \vert }^{p(x,t)}}} \biggr] \end{aligned} \\ &\quad\le \frac{{a(v)v}}{{{2^{{p^{+} } - 1}}{w^{\alpha }}}}{ \vert { \nabla w} \vert ^{p(x,t)}} + {C_{1}}\bigl(\delta,{d_{0}},K,{p^{\pm }}\bigr){ \vert u \vert ^{p'(x,t)}}. \end{aligned}$$

(3.10)

Substituting (3.10) into $J_{3}$ and combining it with $J_{2}$, we obtain

$$\begin{aligned} {J_{3}} \le \frac{1}{2}{J_{2}} + C \int { \int _{{Q_{T}}} {{{ \vert { \nabla u} \vert }^{p(x,t) - 2}} {\, \mathrm{d}}x{\,\mathrm{d}}t} }. \end{aligned}$$

(3.11)

Recall that $0 < \gamma \le {d_{0}}$, $u \in W({Q_{T}}) \cap {L^{\infty }}(0,T; {L^{\infty }}( \Omega ))$. Then, we have

$$\begin{aligned} {J_{4}} \le \int { \int _{{Q_{T}}} {{u^{\sigma }}|\nabla u{|^{p(x,t)}} { \, \mathrm{d}}x{\,\mathrm{d}}t} } \le C \int { \int _{{Q_{T}}} {|\nabla u{|^{p(x,t)}} { \,\mathrm{d}}x{\, \mathrm{d}}t} }, \end{aligned}$$

(3.12)

where C is a positive constant. Thus, we insert the above estimates (3.8), (3.9), (3.11), and (3.12) into (3.7) and dropping the nonnegative terms, we arrive at

$$\begin{aligned} (\delta - 2\varepsilon ) \bigl(1 - {2^{1 - \alpha }} \bigr){\varepsilon ^{1 - \alpha }}\mu (\Omega ) < C. \end{aligned}$$

(3.13)

Secondly, we consider the case $1 < {p^{-} } \le p(x,t) < 2, {p^{+} } \ge 2$. According to the second inequality of Lemma 3.1, it is easily seen that the following inequalities hold

$$\begin{aligned} \begin{aligned} & \bigl\vert {\bigl[a(u)u - a(v)v \bigr]{w^{ - \alpha }} {{ \vert {\nabla w} \vert }^{p(x,t) - 2}}\nabla u \nabla w} \bigr\vert \\ &\quad= \biggl\vert { \biggl[ {(\delta + 1)w \int _{0}^{1} {{{\bigl(\theta u + (1 - \theta )v \bigr)}^{\sigma }}\,\mathrm{d}\theta } + {d_{0}}(u - v)} \biggr]|w{|^{2 - \alpha }} {{ \vert {\nabla u} \vert }^{p(x,t) - 2}}\nabla u \nabla w} \biggr\vert \\ &\quad\le \frac{{a(v)v({p^{ - 1}} - 1)}}{{2{w^{\alpha }}}}{\bigl( \vert { \nabla u} \vert + \vert {\nabla v} \vert \bigr)^{p(x,t)}} { \vert { \nabla {\mathrm{{w}}}} \vert ^{2}} + {C_{1}}\bigl(\delta,{d_{0}},K,{p^{\pm }} \bigr){ \vert w \vert ^{2 - \alpha }} {\bigl( \vert {\nabla u} \vert + \vert {\nabla v} \vert \bigr)^{p(x,t)}} \\ &\quad\le \frac{{a(v)v({p^{ - 1}} - 1)}}{{2{w^{\alpha }}}}{\bigl( \vert { \nabla u} \vert + \vert {\nabla v} \vert \bigr)^{p(x,t)}} { \vert { \nabla {\mathrm{{w}}}} \vert ^{2}} + {C_{1}}\bigl(\delta,{d_{0}},K,{p^{\pm }} \bigr){\bigl( \vert {\nabla u} \vert + \vert {\nabla v} \vert \bigr)^{p(x,t)}}. \end{aligned} \end{aligned}$$

Substituting the above inequality into $J_{3}$, we obtain

$$\begin{aligned} {J_{3}} \le \frac{1}{2}{J_{2}} + C \int { \int _{{Q_{T}}} {{{ \bigl( \vert { \nabla u} \vert + \vert { \nabla v} \vert \bigr)}^{p(x,t) - 2}} {\,\mathrm{d}}x{ \,\mathrm{d}}t} }. \end{aligned}$$

(3.14)

Similar to the case ${p^{-} } \ge 2$, estimate (3.13) still holds using (3.14) instead of (3.11). Note that $\mathop {{\mathrm{{lim}}}} _{ \varepsilon \to 0} (\delta - {\mathrm{{ 2}}}\varepsilon )({\mathrm{{1 }}} - {{\mathrm{{2}}}^{{\mathrm{{1}}} - \alpha }}){\varepsilon ^{{\mathrm{{1}}} - \alpha }}\mu ({\Omega _{\delta }}) = + \infty$, we obtain a contradiction. This means $\mu ({\Omega _{\delta }}) = 0$ and $w \le 0$ a.e. in ${{\mathrm{{Q}}}_{\tau }}$. □

Lemma 3.3

Let ${u_{\varepsilon }}$ be weak solutions of (3.1). Then,

$$\begin{aligned} &{u_{0\varepsilon }} \le {u_{\varepsilon }} \le { \vert {{u_{0}}} \vert _{\infty }} + \varepsilon, \end{aligned}$$

(3.15)

$$\begin{aligned} &{u_{{\varepsilon _{1}}}} \le {u_{{\varepsilon _{2}}}} \quad\textit{for } { \varepsilon _{1}} \le {\varepsilon _{2}}, \end{aligned}$$

(3.16)

where $|{u_{0}}{|_{\infty }} = \mathop {\sup } _{x \in \Omega } |{u_{0}}(x)|$.

Proof

First, we prove ${u_{\varepsilon }} \ge {u_{0\varepsilon }}$ by contradiction. Assume ${u_{\varepsilon }} \le {u_{0\varepsilon }}$ in $Q_{T}^{0}$, $Q_{T}^{0} \subset Q_{T}$. Noting ${u_{\varepsilon }} \ge {u_{0\varepsilon }}$ on $\partial {Q_{T}}$, we may assume that ${u_{\varepsilon }} = {u_{0\varepsilon }}$ on $\partial {Q_{T}}$. With (3.1) and letting $t = 0$, it is easy to see that

$$\begin{aligned} &L{u_{0,\varepsilon }} = - {\beta _{\varepsilon }}({u_{0,\varepsilon }} - {u_{0,\varepsilon }}) = 1, \end{aligned}$$

(3.17)

$$\begin{aligned} &L{u_{\varepsilon }} = - {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0, \varepsilon }}) \le 1. \end{aligned}$$

(3.18)

From Lemma 3.2, we arrive at

$$\begin{aligned} {u_{\varepsilon }}(x,t) \ge {u_{0,\varepsilon }}(x) \quad\text{for any } (x,t) \in {Q_{T}}. \end{aligned}$$

(3.19)

Therefore, we obtain a contradiction.

Secondly, we pay attention to ${u_{\varepsilon }}(t,x) \le { \vert {{u_{0}}} \vert _{\infty }} + \varepsilon$. Applying the definition of ${\beta _{\varepsilon }}( \cdot )$, we have that

$$\begin{aligned} L \bigl({ \vert {{u_{0}}} \vert _{\infty }} + \varepsilon \bigr) = 0,\qquad L{u_{ \varepsilon }} = - {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0, \varepsilon }}) \ge 0. \end{aligned}$$

(3.20)

From (3.20), we obtain

$$\begin{aligned} {u_{\varepsilon }}(t,x) \le { \vert {{u_{0}}} \vert _{\infty }} + \varepsilon \quad\text{on } \partial \Omega \times (0, \mathrm{T}) \end{aligned}$$

(3.21)

and ${u_{\varepsilon }}(t,x) \le { \vert {{u_{0}}} \vert _{\infty }} + \varepsilon$ in Ω. Thus, combining (3.20) and (3.21) and repeating Lemma 3.2, we have

$$\begin{aligned} {u_{\varepsilon }}(t,x) \le { \vert {{u_{0}}} \vert _{\infty }} + \varepsilon\quad \text{in } {Q_{T}}. \end{aligned}$$

(3.22)

Thirdly, we aim to prove (3.16). From (3.1),

$$\begin{aligned} &L{u_{{\varepsilon _{1}}}} = {\beta _{{\varepsilon _{1}}}}({u_{{ \varepsilon _{1}}}} - {u_{0,{\varepsilon _{1}}}}), \end{aligned}$$

(3.23)

$$\begin{aligned} &L{u_{{\varepsilon _{2}}}} = {\beta _{{\varepsilon _{2}}}}({u_{{ \varepsilon _{2}}}} - {u_{0,{\varepsilon _{2}}}}). \end{aligned}$$

(3.24)

It follows by ${\varepsilon _{1}} \le {\varepsilon _{2}}$ and the definition of ${\beta _{\varepsilon }}( \cdot )$ that

$$\begin{aligned} \begin{aligned} &L{u_{0,{\varepsilon _{2}}}} + {\beta _{{\varepsilon _{1}}}}({u_{{ \varepsilon _{2}}}} - {u_{0,\varepsilon }}) \\ &\quad= {\beta _{{\varepsilon _{2}}}}({u_{{\varepsilon _{2}}}} - {u_{0, \varepsilon }}) - {\beta _{{\varepsilon _{1}}}}({u_{{\varepsilon _{1}}}} - {u_{0,\varepsilon }}) \\ &\quad= {\beta _{{\varepsilon _{2}}}}({u_{{\varepsilon _{2}}}} - {u_{0, \varepsilon }}) - {\beta _{{\varepsilon _{1}}}}({u_{{\varepsilon _{2}}}} - {u_{0,\varepsilon }}) \ge 0. \end{aligned} \end{aligned}$$

(3.25)

Thus, combining the initial and boundary conditions in (3.1) can be proved by Lemma 3.2. □

To prove this theorem, we need the following lemmas.

Lemma 3.4

The solution of problem (3.1) satisfies the estimate

$$\begin{aligned} { \Vert {{u_{\varepsilon }}} \Vert _{\infty,{Q_{T}}}} \le { \Vert {{u_{0}}} \Vert _{\infty,\Omega }} + \int _{0}^{T} {{{ \bigl\Vert {f(x,t)} \bigr\Vert }_{\infty,\Omega }} {\,\mathrm{d}}t} + \vert \Omega \vert \cdot T = K(T) < \infty. \end{aligned}$$

Proof

Let us introduce the following function

$$\begin{aligned} {u_{\varepsilon,M}} = \textstyle\begin{cases} M & \text{if }{u_{\varepsilon }} > M, \\ {u_{\varepsilon }}& \text{if } \vert {{u_{\varepsilon }}} \vert < M, \\ - M & \text{if }{u_{\varepsilon }} < - M. \end{cases}\displaystyle \end{aligned}$$

(3.26)

The function $u_{\varepsilon,M}^{2k - 1}$, with $k \in N$, can be chosen as a test function in (3.4). Let ${t_{2} } = t + h,{t_{1}} = t$ in (3.4), with $t,t + h \in ( 0, T )$. Then,

$$\begin{aligned} \begin{aligned} &\frac{1}{{2k}} \int _{t}^{t + h} {\frac{{\mathrm{d}}}{{{\mathrm{d}}t}} \biggl( { \int _{\Omega }{u_{\varepsilon,M}^{2k}{\,\mathrm{d}}x} } \biggr)} {\,\mathrm{d}}t \\ &\qquad{}+ (2k - 1) \int _{t}^{t + h} { \int _{\Omega }{{a_{\varepsilon,M}}({u_{ \varepsilon,M}})u_{\varepsilon,M}^{2(k - 1)}{{ \vert {\nabla {u_{ \varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x{\, \mathrm{d}}t} } \\ &\qquad{}+ \int _{t}^{t + h} { \int _{\Omega }{\bigl[{a_{\varepsilon,M}}({u_{ \varepsilon,M}}) - \gamma \bigr] \cdot u_{\varepsilon,M}^{2k - 1}{{ \vert {\nabla {u_{\varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x{ \,\mathrm{d}}t} } \\ &\quad= \int _{t}^{t + h} { \int _{\Omega }{ \bigl( {f(x,t) - {\beta _{ \varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr) \cdot u_{ \varepsilon,M}^{2k - 1}{\, \mathrm{d}}x{\,\mathrm{d}}t} }. \end{aligned} \end{aligned}$$

(3.27)

Dividing the last equality by h, letting $h \to 0$, and applying Lebesgue’s dominated convergence theorem, we have that

$$\begin{aligned} \begin{aligned} &\frac{1}{{2k}}\frac{{\mathrm{d}}}{{{\mathrm{d}}t}} \int _{\Omega }{u_{ \varepsilon,M}^{2k}{\,\mathrm{d}}x} + (2k - 1) \int _{\Omega }{{a_{ \varepsilon,M}}({u_{\varepsilon,M}})u_{\varepsilon,M}^{2(k - 1)}{{ \vert {\nabla {u_{\varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x} \\ &\qquad{}+ \int _{\Omega }{\bigl[{a_{\varepsilon,M}}({u_{\varepsilon,M}}) - \gamma \bigr] \cdot u_{\varepsilon,M}^{2k - 1}{{ \vert {\nabla {u_{ \varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} \\ &\qquad{}+ \int _{\Omega }{\bigl[{a_{\varepsilon,M}}({u_{\varepsilon,M}}) - \gamma \bigr]u_{\varepsilon,M}^{2k}{{ \vert {\nabla {u_{\varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x} \\ &\quad= \int _{\Omega }{ \bigl( {f(x,t) - {\beta _{\varepsilon }}({u_{ \varepsilon }} - {u_{0}})} \bigr) \cdot u_{\varepsilon,M}^{2k - 1}{ \, \mathrm{d}}x}. \end{aligned} \end{aligned}$$

(3.28)

By Holder’s inequality, we have

$$\begin{aligned} \biggl\vert { \int _{\Omega }{ \bigl( {f(x,t) - {\beta _{\varepsilon }}({u_{ \varepsilon }} - {u_{0}})} \bigr) \cdot u_{\varepsilon,M}^{2k - 1}{ \, \mathrm{d}}x} } \biggr\vert \le \bigl\Vert {u_{\varepsilon,M}} \bigr\Vert _{2k, \Omega }^{2k - 1} \cdot \bigl\Vert {f( \cdot,t) - { \beta _{ \varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr\Vert _{2k,\Omega }. \end{aligned}$$

(3.29)

Using Minkowski’s inequality, we arrive at

$$\begin{aligned} \bigl\Vert {f( \cdot,t) - {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr\Vert _{2k,\Omega } \le \bigl\Vert {f( \cdot,t)} \bigr\Vert _{2k, \Omega } + \bigl\Vert {{\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr\Vert _{2k,\Omega }. \end{aligned}$$

From (3.15) and the definition of ${\beta _{\varepsilon }}( \cdot )$, we have that

$$\begin{aligned} \bigl\Vert {f( \cdot,t) - {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr\Vert _{2k,\Omega } \le \bigl\Vert {f( \cdot,t)} \bigr\Vert _{2k, \Omega } + \vert \Omega \vert . \end{aligned}$$

(3.30)

Recall that $0 < \gamma < {d_{0}}$. Then, we use Lemma 3.1 to find

$$\begin{aligned} \int _{\Omega }{ \bigl[{a_{\varepsilon,M}}({u_{\varepsilon,M}}) - \gamma \bigr] \cdot u_{\varepsilon,M}^{2k - 1}{{ \vert {\nabla {u_{\varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} \ge 0. \end{aligned}$$

(3.31)

Substituting (3.29) and (3.30) into (3.28), we arrive at the inequality

$$\begin{aligned} \begin{aligned} &\bigl\Vert {u_{\varepsilon,M}} \bigr\Vert _{2k,\Omega }^{2k - 1} \frac{{\mathrm{d}}}{{{\mathrm{d}}t}} \bigl\Vert {u_{\varepsilon,M}} \bigr\Vert _{2k,\Omega } + (2k - 1) \int _{\Omega }{{a_{\varepsilon,M}}({u_{ \varepsilon,M}})u_{\varepsilon,M}^{2(k - 1)}{{ \vert {\nabla {u_{ \varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x} \\ &\qquad{}+ \int _{\Omega }{\bigl[{a_{\varepsilon,M}}({u_{\varepsilon,M}}) - \gamma \bigr] \cdot u_{\varepsilon,M}^{2k - 1}{{ \vert {\nabla {u_{ \varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x} \\ &\quad\le \bigl\Vert {u_{\varepsilon,M}} \bigr\Vert _{2k,\Omega }^{2k - 1} \cdot \bigl\Vert {f( \cdot,t)} \bigr\Vert _{2k,\Omega } + \bigl\Vert {u_{ \varepsilon,M}} \bigr\Vert _{2k,\Omega }^{2k - 1} \cdot \vert \Omega \vert . \end{aligned} \end{aligned}$$

(3.32)

Integrating over $(0, t)$ in (3.32) and dropping the nonnegative term (3.31), we arrive at

$$\begin{aligned} \bigl\Vert {u_{\varepsilon,M}} \bigr\Vert _{2k,\Omega } \le \bigl\Vert {u_{\varepsilon,M}( \cdot,0)} \bigr\Vert _{2k,\Omega } + \int _{0}^{T} { \bigl\Vert {f( \cdot,t)} \bigr\Vert _{2k,\Omega }{\,\mathrm{d}}t} + \vert \Omega \vert \cdot T,\quad\forall k \in N. \end{aligned}$$

Then, as $k \to \infty$, we have that

$$\begin{aligned} \Vert {u_{\varepsilon,M}} \Vert _{\infty,\Omega } \le \bigl\Vert {u_{\varepsilon,M}( \cdot,0)} \bigr\Vert _{\infty,\Omega } + \int _{0}^{T} { \bigl\Vert {f( \cdot,t)} \bigr\Vert _{\infty,\Omega }{ \,\mathrm{d}}t} + \vert \Omega \vert \cdot T = K(T). \end{aligned}$$

(3.33)

If we choose $M > K(T)$, then

$$\begin{aligned} {u_{\varepsilon,M}}( \cdot,t) \le \sup \bigl\vert {{u_{\varepsilon,M}}( \cdot,t)} \bigr\vert \le K ( T ) < M \end{aligned}$$

and therefore ${u_{\varepsilon,M}}( \cdot,t) = {u_{\varepsilon }}( \cdot,t)$. □

Lemma 3.5

The solution of problem (3.1) satisfies the estimates

$$\begin{aligned} &\int { \int _{{Q_{T}}} {a \bigl(u_{\varepsilon } \bigr){{ \bigl\vert {\nabla u_{ \varepsilon }} \bigr\vert }^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} } \le K(T){ \vert \Omega \vert ^{\frac{1}{2}}}, \end{aligned}$$

(3.34)

$$\begin{aligned} &{d_{0}} \int { \int _{{Q_{T}}} {{{ \bigl\vert {\nabla u_{\varepsilon }} \bigr\vert }^{p(x,t)}} {\,\mathrm{d}}x{\,\mathrm{d}}t} } \le K(T){ \vert \Omega \vert ^{\frac{1}{2}}}, \end{aligned}$$

(3.35)

$$\begin{aligned} &\int { \int _{{Q_{T}}} {u_{\varepsilon }^{\sigma }{{ \bigl\vert { \nabla u_{ \varepsilon }} \bigr\vert }^{p(x,t)}} {\, \mathrm{d}}x{\,\mathrm{d}}t} } \le K(T){ \vert \Omega \vert ^{\frac{1}{2}}}. \end{aligned}$$

(3.36)

Proof

To prove Lemma 3.5, we proceed as in the proof of Lemma 3.4, and in (3.27) we take $k=1$. We then obtain

$$\begin{aligned} \begin{aligned} &\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}\bigl( \bigl\Vert {u_{\varepsilon }( \cdot ,t)} \bigr\Vert _{2,\Omega } \bigr) + \int _{\Omega }{{a_{\varepsilon,M}}({u_{ \varepsilon }}){{ \bigl\vert {\nabla u_{\varepsilon }} \bigr\vert }^{p(x,t)}} { \,\mathrm{d}}x} \\ &\qquad{}+ \int _{\Omega }{\bigl[{a_{\varepsilon,M}}({u_{\varepsilon,M}}) - \gamma \bigr] \cdot u_{\varepsilon,M}^{2k - 1}{{ \vert {\nabla {u_{ \varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x} \\ &\quad\le \bigl\Vert {f - {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}})} \bigr\Vert _{2,\Omega }. \end{aligned} \end{aligned}$$

Therefore, integrating in time over $(0,t)$, $\forall t \in (0,T)$,

$$\begin{aligned} \begin{aligned} & \bigl\Vert {u_{\varepsilon }( \cdot,t)} \bigr\Vert _{2,\Omega } + \int _{0}^{t} { \int _{\Omega }{{a_{\varepsilon,M}}({u_{ \varepsilon }}){{ \vert { \nabla u_{\varepsilon }} \vert }^{p(x,t)}} { \,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\qquad{}+ \int _{\Omega }{\bigl[{a_{\varepsilon,M}}({u_{\varepsilon,M}}) - \gamma \bigr] \cdot u_{\varepsilon,M}^{2k - 1}{{ \vert {\nabla {u_{ \varepsilon,M}}} \vert }^{p(x,t)}} {\,\mathrm{d}}x} \\ &\quad\le \int _{0}^{T} { \bigl\Vert {f - {\beta _{\varepsilon }}({u_{ \varepsilon }} - {u_{0}})} \bigr\Vert _{2,\Omega }{\,\mathrm{d}}t} \end{aligned} \end{aligned}$$

and since the first and third terms on the left-hand side are nonnegative and recalling the L2-norm

$$\begin{aligned} \int { \int _{{Q_{T}}} {a({u_{\varepsilon }}){{ \bigl\vert {\nabla u_{ \varepsilon }} \bigr\vert }^{p(x,t)}} {\,\mathrm{d}}x{\, \mathrm{d}}t} } \le K(T){ \vert \Omega \vert ^{\frac{1}{2}}}. \end{aligned}$$

(3.37)

From this we obtain (3.34). Since $a({u_{\varepsilon }}) \ge {d_{0}}$, ${a_{\varepsilon,M}}({u_{ \varepsilon }}) \ge u_{\varepsilon }^{\sigma }$, (3.35), and (3.35) are immediate consequences of (3.34). □

Lemma 3.6

The solution of problem (3.1) satisfies the estimate

$$\begin{aligned} \bigl\Vert {u_{\varepsilon t}} \bigr\Vert _{W'({Q_{T}})} \le C \bigl( \sigma,{p^{\pm }},K(T), \vert \Omega \vert \bigr). \end{aligned}$$

Proof

From identity (3.5), we obtain

$$\begin{aligned} \begin{aligned} &\int { \int _{{Q_{T}}} {u_{\varepsilon t}\xi {\,\mathrm{d}}x{ \,\mathrm{d}}t} } \\ &\quad= - \int { \int _{{Q_{T}}} {a({u_{\varepsilon }}){{ \vert {\nabla {u_{ \varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{\varepsilon }} \nabla \xi {\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\qquad{}- \int { \int _{{Q_{T}}} {\bigl[a({u_{\varepsilon }}) - \gamma \bigr]{{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t)}}\xi { \,\mathrm{d}}x{ \,\mathrm{d}}t} } + \int { \int _{{Q_{T}}} {f \cdot \xi {\,\mathrm{d}}x{ \,\mathrm{d}}t} } \\ &\quad= - {A_{1}} - {A_{1}} + {A_{1}}, \end{aligned} \end{aligned}$$

(3.38)

where

$$\begin{aligned} &{A_{1}} = \int { \int _{{Q_{T}}} {a({u_{\varepsilon }}){{ \vert { \nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{ \varepsilon }}\nabla \xi { \,\mathrm{d}}x{\,\mathrm{d}}t} },\\ &{A_{2}} = \int { \int _{{Q_{T}}} { \bigl[a({u_{\varepsilon }}) - \gamma \bigr]{{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t)}}\xi {\,\mathrm{d}}x{ \,\mathrm{d}}t} },\qquad {A_{3}} = \int { \int _{{Q_{T}}} {f \cdot \xi {\,\mathrm{d}}x{ \,\mathrm{d}}t} }. \end{aligned}$$

First, we pay attention to $A_{1}$. Using Holder inequalities we obtain

$$\begin{aligned} \begin{aligned} \vert {{A_{1}}} \vert & \le \int _{0}^{t} { \int _{\Omega }{a({u_{ \varepsilon }}){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 1}} \vert {\nabla \xi } \vert {\, \mathrm{d}}x{\,\mathrm{d}}t} } \\ &\le 2{ \bigl\Vert {a({u_{\varepsilon }}){{ \vert {\nabla {u_{ \varepsilon }}} \vert }^{p(x,t) - 1}}} \bigr\Vert _{p'(x,t)}} { \Vert {\nabla \xi } \Vert _{p(x,t)}}. \end{aligned} \end{aligned}$$

When $\int _{0}^{t} {\int _{\Omega }{{{ ( {a({u_{\varepsilon }}){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 1}}} )}^{ \frac{{p(x,t)}}{{p(x,t) - 1}}}}{\,\mathrm{d}}x{\,\mathrm{d}}t} } \ge 1$, we arrive at

$$\begin{aligned} \vert {{A_{1}}} \vert \le 2{ \biggl( { \int _{0}^{t} { \int _{ \Omega }{{{ \bigl( {a({u_{\varepsilon }}){{ \vert {\nabla {u_{ \varepsilon }}} \vert }^{p(x,t) - 1}}} \bigr)}^{ \frac{{p(x,t)}}{{p(x,t) - 1}}}} {\, \mathrm{d}}x{\,\mathrm{d}}t} } } \biggr)^{ \frac{1}{{{{p'}^{+} }}}}} \cdot { \Vert {\nabla \xi } \Vert _{p(x,t)}}. \end{aligned}$$

(3.39)

Moreover, when $\int _{0}^{t} {\int _{\Omega }{{{ ( {a({u_{\varepsilon }}){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 1}}} )}^{ \frac{{p(x,t)}}{{p(x,t) - 1}}}}{\,\mathrm{d}}x{\,\mathrm{d}}t} } < 1$, we obtain

$$\begin{aligned} \vert {{A_{1}}} \vert \le 2{ \biggl( { \int _{0}^{t} { \int _{ \Omega }{{{ \bigl( {a({u_{\varepsilon }}){{ \vert {\nabla {u_{ \varepsilon }}} \vert }^{p(x,t) - 1}}} \bigr)}^{ \frac{{p(x,t)}}{{p(x,t) - 1}}}} {\, \mathrm{d}}x{\,\mathrm{d}}t} } } \biggr)^{ \frac{1}{{{{p'}^{-} }}}}} \cdot { \Vert {\nabla \xi } \Vert _{p(x,t)}}. \end{aligned}$$

(3.40)

Combining (3.39) and (3.40), and using Lemma 3.5, we arrive at

$$\begin{aligned} \vert {{A_{1}}} \vert \le { \bigl( {2 \bigl[{{ \bigl({K^{2}}(T) + 1 \bigr)}^{ \sigma /2}} + {d_{0}} \bigr]} \bigr)^{\frac{1}{{{p^{\pm }} - 1}}}} {K^{2}}(T) \vert \Omega \vert \cdot { \Vert \xi \Vert _{W({Q_{T}})}}. \end{aligned}$$

(3.41)

Secondly, we calculate $A_{2}$ and $A_{3}$. Following a similar procedure as (3.41), we obtain

$$\begin{aligned} &\begin{aligned} \vert {{A_{2}}} \vert \le{}& 2{\bigl[{\bigl({K^{2}}(T) + 1\bigr)^{\sigma /2}} + {d_{0}}\bigr]^{ \frac{1}{{{p^{\pm }} - 1}}}} {K^{2}}(T) \vert \Omega \vert \cdot { \Vert \xi \Vert _{W({Q_{T}})}} \\ &{}+ 2{\gamma ^{\frac{1}{{{p^{\pm }} - 1}}}} {K^{2}}(T) \vert \Omega \vert \cdot { \Vert \xi \Vert _{W({Q_{T}})}}, \end{aligned} \end{aligned}$$

(3.42)

$$\begin{aligned} &\vert {{A_{3}}} \vert \le 2{ \vert f \vert _{\infty }} \vert T \vert \cdot { \Vert \xi \Vert _{W({Q_{T}})}}. \end{aligned}$$

(3.43)

Substituting (3.41), (3.42), and (3.43) into (3.38), we conclude that

$$\begin{aligned} \begin{aligned} \int { \int _{{Q_{T}}} {u_{\varepsilon t}\xi {\,\mathrm{d}}x{ \,\mathrm{d}}t} } \le {}&4{\bigl[{\bigl({K^{2}}(T) + 1\bigr)^{\sigma /2}} + {d_{0}}\bigr]^{ \frac{1}{{{p^{\pm }} - 1}}}} {K^{2}}(T) \vert \Omega \vert \cdot { \Vert \xi \Vert _{W({Q_{T}})}} \\ &{}+ 2{\gamma ^{\frac{1}{{{p^{\pm }} - 1}}}} {K^{2}}(T) \vert \Omega \vert \cdot + 2{ \vert f \vert _{\infty }} \vert T \vert \cdot { \Vert \xi \Vert _{W({Q_{T}})}}. \end{aligned} \end{aligned}$$

Then, we obtain Lemma 3.6. □

4 Proof of the main results

In this section, we are ready to prove Theorem 2.1 and Theorem 2.2. From (3.15), Lemma 3.5, and Lemma 3.6, we see that ${u_{\varepsilon}}$ is bounded and increasing in ε, which implies the existence of a function u, such that, as $\varepsilon \to 0$

$$\begin{aligned} &{u_{\varepsilon }} \to u \quad\text{a.e. in } {\Omega _{T}} , \end{aligned}$$

(4.1)

$$\begin{aligned} &\nabla {u_{\varepsilon }} \to \nabla u\quad \text{weakly in } {L^{p(x,t)}}({Q_{T}}), \end{aligned}$$

(4.2)

$$\begin{aligned} &\frac{\partial }{{\partial t}}{u_{\varepsilon }} \to \frac{\partial }{{\partial t}}u \quad\text{weakly in }W'({Q_{T}}), \end{aligned}$$

(4.3)

$$\begin{aligned} &{\mathrm{{a}}}({u_{\varepsilon }}){ \vert {\nabla {u_{\varepsilon }}} \vert ^{p(x,t) - 2}} {D_{i}} {u_{\varepsilon }} \to {A_{i}}(x,t)\quad \text{weakly in } {L^{p'(x,t)}}({Q_{T}}), \end{aligned}$$

(4.4)

$$\begin{aligned} &{ \vert {\nabla {u_{\varepsilon }}} \vert ^{p(x,t) - 2}} {D_{i}} {u_{ \varepsilon }} \to {W_{i}}(x,t) \quad\text{weakly in } {L^{p'(x,t)}}({Q_{T}}), \end{aligned}$$

(4.5)

for some functions $u \in W({Q_{T}})$, ${A_{i}}(x,t) \in {L^{p'(x,t)}}({Q_{T}})$, ${W_{i}}(x,t) \in {L^{p'(x,t)}}({Q_{T}})$.

Lemma 4.1

For almost all $(x,t) \in {Q_{T}}$,

$$\begin{aligned} {A_{i}}(x,t) = a(u) {W_{i}}(x,t),\quad i = 1,2, \ldots,N. \end{aligned}$$

Proof

In (4.4) and (4.5), letting $\varepsilon \to 0$, we have that

$$\begin{aligned} &\int { \int _{{Q_{T}}} {a(u_{\varepsilon }){{ \vert {\nabla {u_{ \varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{\varepsilon }} \nabla \xi {\,\mathrm{d}}x{\,\mathrm{d}}t} } = \sum_{i} { \int { \int _{{Q_{T}}} {{A_{i}}(x,t) \cdot {D_{i}} \xi {\,\mathrm{d}}x{\,\mathrm{d}}t} } }, \end{aligned}$$

(4.6)

$$\begin{aligned} &\int { \int _{{Q_{T}}} {{{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{\varepsilon }}\nabla \xi {\,\mathrm{d}}x{\, \mathrm{d}}t} } = \sum_{i} { \int { \int _{{Q_{T}}} {{W_{i}}(x,t) \cdot {D_{i}} \xi {\,\mathrm{d}}x{\,\mathrm{d}}t} } }. \end{aligned}$$

(4.7)

By Lebesgue’s dominated convergence theorem, we have

$$\begin{aligned} \mathop {\lim } _{\varepsilon \to 0} \int { \int _{{Q_{T}}} { \bigl[a \bigl(u_{ \varepsilon } \bigr) - a(u) \bigr]{A_{i}}(x,t) \cdot {D_{i}}\xi {\, \mathrm{d}}x{\,\mathrm{d}}t} } = 0. \end{aligned}$$

(4.8)

Hence, we have

$$\begin{aligned} \begin{aligned} &\mathop {\lim } _{\varepsilon \to 0} \int { \int _{{Q_{T}}} {a\bigl(u_{ \varepsilon }\bigr){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}} {D_{i}} {u_{\varepsilon }} {D_{i}}\xi - a(u){W_{i}}(x,t){D_{i}} \xi {\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\quad= \mathop {\lim } _{\varepsilon \to 0} \int { \int _{{Q_{T}}} {\bigl[a\bigl(u_{ \varepsilon }\bigr) - a(u)\bigr] \cdot {{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}} {D_{i}} {u_{\varepsilon }} {D_{i}}\xi { \,\mathrm{d}}x{ \,\mathrm{d}}t} } \\ &\qquad{}+ \mathop {\lim } _{\varepsilon \to 0} \int { \int _{{Q_{T}}} {a(u) \bigl( {{{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}} {D_{i}} {u_{\varepsilon }} - {W_{i}}(x,t)} \bigr){D_{i}}\xi {\,\mathrm{d}}x{ \, \mathrm{d}}t} } = 0. \end{aligned} \end{aligned}$$

This completes the proof of Lemma 4.1. □

Lemma 4.2

For almost all $(x,t) \in {Q_{T}}$,

$$\begin{aligned} {W_{i}}(x,t) = |\nabla u{|^{p(x,t) - 2}} {D_{i}}u,\quad i = 1,2, \ldots,N. \end{aligned}$$

Proof

In (3.5), choosing $\xi = \Phi \cdot ({u_{\varepsilon }} - u)$ with $\Phi \in W({Q_{T}})$, $\Phi \ge 0$, we have

$$\begin{aligned} \begin{aligned} &\int { \int _{{Q_{T}}} {{\partial _{t}}u_{\varepsilon }\cdot (u_{ \varepsilon }- u) \cdot \Phi + \Phi \cdot a\bigl(u_{\varepsilon } \bigr){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{ \varepsilon }}\nabla ({u_{\varepsilon }} - u){\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &\quad{}+ \int { \int _{{Q_{T}}} {\bigl(u_{\varepsilon } - u\bigr) \cdot a\bigl(u_{ \varepsilon }\bigr){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{\varepsilon }}\nabla \Phi - f(x,t) (u_{\varepsilon }- u){ \,\mathrm{d}}x{\,\mathrm{d}}t} } = 0. \end{aligned} \end{aligned}$$

It follows that

$$\begin{aligned} \mathop {\lim } _{\varepsilon \to \infty } \int { \int _{{Q_{T}}} {\Phi \cdot a \bigl(u_{\varepsilon } \bigr){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{\varepsilon }}\nabla ({u_{ \varepsilon }} - u){\,\mathrm{d}}x{\,\mathrm{d}}t} } = 0. \end{aligned}$$

(4.9)

On the other hand, from ${u_{\varepsilon }},u \in {L^{\infty }}(Q_{T})$, $|\nabla u| \in {L^{p(x,t)}}({Q_{T}})$, we obtain

$$\begin{aligned} &\mathop {\lim } _{\varepsilon \to \infty } \int { \int _{{Q_{T}}} {\Phi \cdot a(u){{ \vert {\nabla {u}} \vert }^{p(x,t) - 2}} \nabla {u}\nabla ({u_{\varepsilon }} - u){\,\mathrm{d}}x{\,\mathrm{d}}t} } = 0, \end{aligned}$$

(4.10)

$$\begin{aligned} &\mathop {\lim } _{\varepsilon \to \infty } \int { \int _{{Q_{T}}} {\Phi \cdot \bigl[a \bigl(u_{\varepsilon } \bigr) - a(u) \bigr]{{ \vert {\nabla {u}} \vert }^{p(x,t) - 2}} \nabla {u}\nabla ({u_{\varepsilon }} - u){ \, \mathrm{d}}x{\, \mathrm{d}}t} } = 0. \end{aligned}$$

(4.11)

Note that

$$\begin{aligned} \begin{aligned} 0 \le{}& \bigl( {{{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{\varepsilon }} - {{ \vert {\nabla u} \vert }^{p(x,t) - 2}}\nabla u} \bigr) \cdot ( {\nabla {u_{\varepsilon }} - \nabla u} ) \\ \le{}& \frac{1}{{{d_{0}}}} \bigl[ {a\bigl(u_{\varepsilon }\bigr){{ \vert { \nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}}\nabla {u_{ \varepsilon }} + \bigl[a\bigl(u_{\varepsilon }\bigr) - a(u) \bigr] \cdot {{ \vert { \nabla u} \vert }^{p(x,t) - 2}}\nabla u} \bigr] \cdot ( { \nabla {u_{\varepsilon }} - \nabla u} ) \\ &{}- \frac{1}{{{d_{0}}}}a(u){ \vert {\nabla u} \vert ^{p(x,t) - 2}}\nabla u \cdot ( {\nabla {u_{\varepsilon }} - \nabla u} ). \end{aligned} \end{aligned}$$

(4.12)

By (4.9)–(4.12), we obtain

$$\begin{aligned} \mathop {\lim } _{\varepsilon \to \infty } \int { \int _{{Q_{T}}} { \bigl( {{{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 2}} \nabla {u_{\varepsilon }} - {{ \vert {\nabla u} \vert }^{p(x,t) - 2}} \nabla u} \bigr) \cdot ( {\nabla {u_{\varepsilon }} - \nabla u} ){\,\mathrm{d}}x{\,\mathrm{d}}t} } = 0. \end{aligned}$$

(4.13)

Then, the proof of Lemma 4.2 is complete. □

Lemma 4.3

As $\varepsilon \to 0$, we have

$$\begin{aligned} {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}}) \to \xi \in G(u - {u_{0}}). \end{aligned}$$

(4.14)

Proof

Using (3.15) and the definition of ${\beta _{\varepsilon }}$, we have

$$\begin{aligned} {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}}) \to \xi \quad \text{as } \varepsilon \to 0. \end{aligned}$$

Now, we prove $\xi \in G(u - {u_{0}})$. According to the definition of $G( \cdot )$, we only need to prove that if $u({x_{0}}, {t_{0}}) > {u_{0}}({x_{0}})$,

$$\begin{aligned} \xi ({x_{0}}, {t_{0}}) = 0. \end{aligned}$$

In fact, if $u({x_{0}},{t_{0}}) > {u_{0}}({x_{0}})$, there exist a constant $\lambda > 0$ and a δ-neighborhood ${B_{\delta }}({x_{0}},{t_{0}})$ such that if ε is small enough, we have

$$\begin{aligned} {u_{\varepsilon }}(x,t) \ge {u_{0}}(x) + \lambda,\quad\forall (x, t) \in {B_{\delta }}({x_{0}},{t_{0}}). \end{aligned}$$

Thus, if ε is small enough, we have

$$\begin{aligned} 0 \ge {\beta _{\varepsilon }}({u_{\varepsilon }} - {u_{0}}) \ge { \beta _{\varepsilon }}(\lambda ) = 0, \quad\forall (x, t) \in {B_{ \delta }}({x_{0}}, {t_{0}}). \end{aligned}$$

Furthermore, it follows by $\varepsilon \rightarrow 0$ that

$$\begin{aligned} \xi (x, t) = 0, \quad\forall (x, t) \in {B_{\delta }}({x_{0}}, {t_{0}}). \end{aligned}$$

Hence, (4.13) holds, and the proof of Lemma 4.3 is complete. □

The proof of Theorem 2.1.

Applying (3.15), (3.16), and Lemma 4.3, it is clear that

$$\begin{aligned} u(x,t) \le {u_{0}}(x), \quad\text{in } {\Omega _{T}},\qquad u(x,0) = {u_{0}}(x), \quad\text{in } \Omega,\qquad\xi \in G(u - {u_{0}}), \end{aligned}$$

thus (a), (b), and (c) hold. The remaining arguments of the existence part are the same as those of Theorem 2.1 in [8] by a standard limiting process. Thus, we omit the details. □

The proof of Theorem 2.2

We argue by contradiction. Suppose $(u,{\xi _{1}})$ and $(v,{\xi _{2}})$ are two nonnegative weak solutions of problem (1.1). Define $w = u - v$,

$$\begin{aligned} F(w) = \textstyle\begin{cases} - \frac{1}{{\alpha - 1}}{w^{1 - \alpha }}& \text{if }w > 0, \\ 0 &\text{if }w \le 0, \end{cases}\displaystyle \end{aligned}$$

(4.15)

and let $\xi = {F_{\varepsilon }}(w) \in Z$ be a test function in (3.4),

$$\begin{aligned} \begin{aligned} 0 \ge{}& \int { \int _{{Q_{T}}} {{w_{t}}F(w) + a(v)v \bigl( { \bigl\vert \nabla u{|^{p(x,t) - 2}}\nabla u - \bigr\vert \nabla v{|^{p(x,t) - 2}}\nabla v} \bigr)\nabla {F_{ \varepsilon }}(w){\,\mathrm{d}}x{\, \mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u)u - a(v)v\bigr]|\nabla u{|^{p(x,t) - 2}} \nabla u\nabla {F_{\varepsilon }}(w){\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u) - \gamma \bigr] \bigl( { \bigl\vert \nabla u{|^{p(x,t)}} - \bigr\vert \nabla v{|^{p(x,t)}}} \bigr){\, \mathrm{d}}x{\,\mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u) - a(v)\bigr]|\nabla u{|^{p(x,t)}} { \,\mathrm{d}}x{ \,\mathrm{d}}t} } - \int { \int _{{Q_{T}}} {({\xi _{1}} - {\xi _{2}})F(w){ \,\mathrm{d}}x{\,\mathrm{d}}t} }. \end{aligned} \end{aligned}$$

(4.16)

Now, we prove

$$\begin{aligned} \int _{\Omega }{({\xi _{1}} - {\xi _{2}})F(w){ \,\mathrm{d}}x{\,\mathrm{d}}t} \le 0. \end{aligned}$$

(4.17)

On the one hand, if ${u_{1}}(x,t) > {u_{2}}(x,t)$, then using (3.16) yields

$$\begin{aligned} {u_{1}}(x,t) > {u_{0}}(x). \end{aligned}$$

(4.18)

From (2.1) and (4.18), it is easy to see that

$$\begin{aligned} {\xi _{1}} = 0 < {\xi _{2}}. \end{aligned}$$

(4.19)

Combining (4.18) and (4.19) and the fact that $\alpha = \frac{1}{2}\sigma > 1$, (4.16) is obtained.

On the other hand, if ${u_{1}}(x,t) < {u_{2}}(x,t)$, it is easy to see that $F(w) = 0$. Equation (4.16) still holds.

Using (4.16) in (4.15) and dropping the nonnegative term, (4.15) becomes

$$\begin{aligned} \begin{aligned} 0 \ge{}& \int { \int _{{Q_{T}}} {{w_{t}}F(w) + a(v)v \bigl( { \bigl\vert \nabla u{|^{p(x,t) - 2}}\nabla u - \bigr\vert \nabla v{|^{p(x,t) - 2}}\nabla v} \bigr)\nabla {F_{ \varepsilon }}(w){\,\mathrm{d}}x{\, \mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u)u - a(v)v\bigr]|\nabla u{|^{p(x,t) - 2}} \nabla u\nabla {F_{\varepsilon }}(w){\,\mathrm{d}}x{\,\mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u) - \gamma \bigr] \bigl( { \bigl\vert \nabla u{|^{p(x,t)}} - \bigr\vert \nabla v{|^{p(x,t)}}} \bigr){\, \mathrm{d}}x{\,\mathrm{d}}t} } \\ &{}+ \int { \int _{{Q_{T}}} {\bigl[a(u) - a(v)\bigr]|\nabla u{|^{p(x,t)}} { \,\mathrm{d}}x{ \,\mathrm{d}}t} }. \end{aligned} \end{aligned}$$

By the above inequality and combining the initial and boundary conditions in (1.1), the uniqueness of the solution can be proved following the similar proof of (3.7)–(3.14). □

5 Conclusion

In this paper, an initial Dirichlet problem of degenerate parabolic variational inequalities in the following form

$$\begin{aligned} \textstyle\begin{cases} \min \{ Lu,u(x,0) - {u_{0}}\} = 0, & (x,t) \in {Q_{T}}, \\ u(x,t) = 0, & (x,t) \in {\Gamma _{T}}, \\ u(x,0) = {u_{0}}, &x \in \Omega, \end{cases}\displaystyle \end{aligned}$$

is studied. The existence and uniqueness of the solutions in the weak sense are proved by the energy method and a limit process. The localization property of weak solutions is also discussed.