Existence of a Sign-Changing Weak Solution to Doubly Nonlinear Parabolic Equations

In this paper, assuming the initial-boundary datum belonging to suitable Sobolev and Lebesgue spaces, we prove the global existence result for a (possibly sign changing) weak solution to the Cauchy–Dirichlet problem for doubly nonlinear parabolic equations of the form ∂t|u|q-1u-Δpu=0inΩ∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _t\left( |u|^{q-1}u\right) -\Delta _p u=0\quad \text {in}\,\,\,\Omega _\infty , \end{aligned}$$\end{document}where p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document} and q>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q>0$$\end{document}. This is a fair improvement of the preceding result by authors (Nonlinear Anal 175C :157–172, 2018). The key tools we employ are energy estimates for approximate equations of Rothe type and the integral strong convergence of gradients of approximate solutions.


Introduction
Let p > 1 and q > 0 be given exponents and be a bounded domain in R n with n 2 with smooth boundary. The aim of this paper is to establish the existence result of solutions of the Cauchy-Dirichlet problem for the doubly nonlinear parabolic type equation of the form (1.1) Here, the unknown function u = u(x, t) is a real-valued function defined on ∞ and p u := div |Du| p−2 Du denotes the p-Laplacian, where by Du = (∂ x i u) 1 i n we mean the spatial gradient of u with respect to x, and the initial datum u 0 is given in some function space. The initial condition u = u 0 in × {0} has to be understood in the L q+1 -sense, while the boundary condition u = 0 on the lateral boundary is given in the usual trace sense; see (D3) in Definition 1 below. The doubly nonlinear parabolic equations (1.1) 1 are classified as fast diffusion equations in the case q > p−1 and slow diffusion equations in the case q < p − 1, whose criterion depends on the expansion of support of the solutions. In the special case q = p − 1, (1.1) 1 covers the Trudinger equation, which has first been studied by Trudinger [32].
Prior to stating our main results, we shall give a brief outlook of the literatures and a motive for (1.1). The existence problem for (1.1) has been already studied in some preceding results. The general form equations of which the prototype equation being (1.1) was firstly treated for the case that q 1 in Alt & Luckhaus [2]. By a variational method the existence of non-negative solutions is succeeded by Bögelein, Duzaar, Marcellini & Scheven [4]. Their approach enables us to extend the result to non-negative variational solutions for more general parabolic equations of the type where f : × R × R n → (−∞, ∞] is a Carathéodory function satisfying certain convexity and coercivity assumptions. The prototype of the integrand f is of the form f = f (x, Du) = α(x)|Du| p + β(x)|Du| q with 1 < p < q and α, β being non-negative coefficients such that α(x) + β(x) ν > 0. The historical notes and the overview of regularity theory for the elliptic equation associated with (1.2) with the nonstandard growth condition similarly as above are addressed in Mingione & Rȃdulescu [25] and references therein. Besides, the existence of variational solutions to the Cauchy-Dirichlet problem with time-dependent boundary values in the case q > 1 is established by Schätzler [30]. We would also like to state that the Hölder regularity of sign-changing weak solutions has been shown in [8] in the case q = p−1. The regularity issue for a weak solution to the doubly nonlinear equation (1.1) will be the challenging problem in our near future. In our previous result by the authors [27], the existence of a solution in the energy class is studied in the case that q + 1 = np n− p with n 3 and 2 p < n. The definition of a weak solution is stated in Definition 1. Our main aim in this paper is to extend and refine [27], showing the global existence of a weak solution of (1.1) for any initial datum of the energy class in the general case that q > 0 and p > 1. We shall show the existence of a weak solution satisfying the energy inequality, of which the proof relies on the same Rothe-type approximation as in [27]. The new content gained here is the strong convergence of the gradients of approximating solutions (Lemma 4.8). The proof uses the exponential mollification (see Lemma 2.1), which is an effective measure to control the power nonlinearity in time derivative, and the so-called boundary integral, devised in [4]. It may have an interest of its own self. See formulae (2.1) and (2.2) for the definition of mollification and boundary integral.
The doubly nonlinear parabolic equations appear in a model of some physical phenomena like plasma physics or turbulent filtration of liquids (see e.g., [31] and references therein for a detailed description) and also describe the gradient flow associated with the Sobolev inequality and is related to the so-called Yamabe flow in differential geometry in the case that p = 2. Also see [28] and references therein for the nonlinear parabolic equations on Riemannian manifolds. Besides, the gradient flow of the p-elastic energy is one of geometric aspects of our motive, see [9] for a detailed description. Our motive for studying the equation (1.1) is the gradient flow associated with the Sobolev inequality in the L p -setting, referred to the p-Sobolev flow. In the Sobolev critical case that q + 1 = np n− p with n 3 and 2 p < n, the global existence of a positive weak solution with the gradient regularity is shown in [19,20], where the so-called intrinsic scaling method peculiar to the doubly nonlinear equation (1.1) is devised to construct the global solution of the p-Sobolev flow. The results and methods presented here will enable us to show the global existence and regularity of the p-Sobolev flow in the general case that q > 0 and p > 1 in our near future work.
Our main result reads as follows: Theorem 1.1 Let p > 1, q > 0 be given and assume that the initial datum u 0 belongs to W 1, p 0 ( ) ∩ L q+1 ( ). Then there exists a global in time weak solution to (1.1) in the sense of Definition 1 fulfilling the following energy structures and regularity: with a constant c ≡ c( p, q).
Our method developed here can be applied for Cauchy-Dirichlet problem (1.1) for doubly nonlinear parabolic equations of more general form, having the variational structure with controllable growth on gradients. Here, unless otherwise specified, we shall deal with the prototype equation (1.1) 1 and present the transparent details of proof.
A synopsis of this paper is in the following: In the next section, we display the notation and summarize auxiliary material used throughout the paper. In Sect. 3, we construct approximate solutions of (1.1). Sect. 4 contains the results of the convergence of approximate solutions of (1.1). In Sect. 5, we complete the proof of our Theorem 1.1 via the weak compactness method. Appendix A is devoted to proving Lemma 3.1 by the Direct Method in the Calculus of Variations. In Appendix B, we prove the finite integration by parts formula (Lemma 4.9). Appendix C is devoted to the proof of the needed lemma for Sect. 5 (Lemma 5.1). In the final Appendix D, we show the initial regularity (5.6).

Preliminaries
We shall split this section in three parts: first, we display our notation, then we list some technical tools. Finally, we present the definition of weak solutions to Eq. (1.1).

Notation
In the course of the paper, we work in the Euclidean space R n in a fixed dimension n 2 and ⊂ R n denotes a bounded domain with the smooth boundary. For T ∈ (0, ∞), let T := × (0, T ) be a space-time cylinder. As is customary, denotes the usual Euclidean ball with radius > 0, centered at x 0 ∈ R n . We adopt the convention of writing B instead of B (x 0 ), when the center is origin 0 of R n , or when the center is clear from the context. Given a measurable set O ⊂ R k and an integrable map w : With O ⊂ R k being a measurable set and w being a measurable function defined on O, we shall use the shorthand notation Finally, we will list the general notation. Let us denote by c, c 1 , c 2 , · · · different positive constants in a given context. Relevant dependencies on parameters will be emphasized using parentheses, e.g., c ≡ c(n, γ, p) means that c depends on n, γ , and p. For the sake of readability, the dependencies of the constants will be often omitted within the chains of estimates. Furthermore, by ( · ) , we mean the -th line of the Eq. ( · ).

Exponential Mollification in Time
In this subsection, we introduce the technique of the exponential mollification in time, which appears in the paper by Landes [23]. Kinnunen & Lindqvist [18] contributed the validity of this technique for the doubly nonlinear PDEs theory. This mollification enables us to overcome the lack of regularity in the time variable. The properties of this mollification are applied to the variant doubly nonlinear equations, as seen in the literatures [3,[6][7][8].
Let ⊂ R n be a bounded domain and 0 < T < ∞. In h ∈ L p ( T ) and the following quantitative estimate holds true: The same statement holds true for and solve the ODE:

Some Preliminary Material
In this subsection, we collect the auxiliary material used throughout the paper. We shall use the boundary integral term, devised by Bögelein et. al [4]: This boundary term is a crucial quantity for our argument. We state the estimates for the boundary term B, whose precise proof is presented in [ and, in particular, when α 2 where the dot · denotes the usual inner product in R k .
We next list the following integral convergence lemma that will be used later, whose proof can be seen in [ (3) there is a function g ∈ L 1 ( T ) such that  g h → g almost everywhere in T , Then there holds that

Weak Solution
We make precise the notion of solution to (1.1) that we will use.

Definition 1 (Weak solution)
Assume that the initial datum u 0 is in the class W and u satisfies the boundary condition in the following sense:

Approximate Solutions
In this section, we will construct approximate solutions of (1.1) satisfying certain energy estimates. Following [27,Sect. 3], we define a family of elliptic equations of Rothe type (as seen in [29]). Let h ∈ (0, 1] be a positive number, sent to zero later. Starting with the initial datum u 0 belonging to W 1, p 0 ( ) ∩ L q+1 ( ), we inductively construct a sequence u i ∈ W 1, p 0 ( ) ∩ L q+1 ( ) of solutions to the following elliptic equations for every i ∈ N: The existence of the weak solution u i to (3.1) is guaranteed by the following lemma. The proof is postponed and will be given in Appendix A.  (3.1) in the following sense: Again, let h < 1 be a fixed positive number, sent to zero later. Set t i := ih for any non-negative integer i. Following the scheme as presented in [16,27], we construct the . For the solutions (u i ) i∈N to (3.1) and i = 1, 2, . . ., we define the piecewise constant functions We call all of six functionsū h ,v h ,w h , u h , v h , and w h as approximate solutions of (1.1). At this stage, (3.1) is rewritten as equivalently, in the sense of distribution, holds for any positive T < ∞ and all testing functions ϕ ∈ L 1 (0, T ; W In the following, we shall make some integral estimates for the approximate solutions. Here, we always suppose the setting as follows : For any positive number T < ∞ and any step size h ∈ (0, 1], let k be the positive integer fulfilling (k − 1)h T < kh.

Lemma 3.2 (Energy estimates) Letū h and u h be the approximate solutions of (1.1) defined as in (3.3)-(3.4). The following energy estimates hold for any positive T < ∞ :
Proof The proof can be achieved by the following arguments in [27, Lemma 3.2]; therefore, we will only sketch it. Testing ξ = u i in (3.2) and summing up on i = 1, . . . , k, subsequently, we use Young's inequality to obtain (3.6). Similarly, . This finishes the proof.
The statement of Lemma 3.2 also holds for the linear interpolation u h as follows.

Lemma 3.3 (Energy estimates) Let u h be the approximate solution of (1.1) defined by (3.4) 1 . Then, the following energy estimates hold true for any positive T < ∞ :
We next deduce the integral bounds of time derivative.

Lemma 3.4 (Time-derivative estimates) Let v h and w h be the approximate solutions of (1.1) defined by (3.4). Then, the time derivatives of approximating solutions
and, furthermore, if q 1 the time derivatives of approximating solutions with a positive constant c ≡ c( p, q).
Proof As before, we choose a positive integer k to satisfy t k−1 T < t k . By (2.5) in We observe that by the very definition (3.4) 3 , (3.12), and (3.7) from Lemma 3.2 which implies (3.10).
As we are considering the case q 1, (2.5) in Lemma 2.3 with α = q + 1 implies and this combined with Hölder's inequality twice with pair of exponents (2, 2) and finishing the proof of (3.11).

Convergence of Approximate Solutions
This section is devoted to deriving the convergence of approximate solution of (1.1) defined in (3.3)-(3.4) by means of the truncated function stated below. This scheme is necessary for the strong convergence of approximate solutions in the space-time Sobolev space; see Lemma 4.4. In the sequel, we assume the initial datum u 0 belongs to W 1, p 0 ( ) ∩ L q+1 ( ). We introduce the following truncated function: For the solution (u i (x)) i∈N to (3.1) and ∈ N 2 , let us define as (4.1) From the definition above,

Energy Estimates for the Truncated Solution
Firstly, we deduce certain energy estimate of the truncated solution defined in (4.1) above.
± be the function defined by (4.1). Then there holds for any whenever q 1 and, Proof We shall follow almost the same argument as in [16,Lemma 4.1] although, for the sake of completeness, we shall nevertheless give the full proof. We confine the case (u h ) ( ) + only since the other case is similarly treated. As before, let a positive integer k to satisfy t k−1 T < t k . To begin, we see by energy estimate (3.7) in Lemma 3.2 that for a constant c ≡ c( p, q). In the sequel, set [u i L] ≡ ∩ {u i L} for short; other symbols are also abbreviated similarly. We shall distinguish between the cases q 1 and 0 < q < 1.
In the first case q 1, for any i = 1, . . . , k, where, in the second line, we discarded the non-negative second term on the righthand side and, further, in the fourth line, we used the fact that . This together with (4.4) implies the first desired estimate (4.2).
Hereafter, it only remains to consider the latter case 0 < q < 1. For this, we need the following quantitative estimate: In order to verify (4.5), a distinction must be made among the cases in the following: Firstly, on the set 1 : = 0 and thus, (4.5) is clearly valid. On the set \ ( 1 ∪ 2 ), we distinguish among the four cases (I)-(IV).
In this region, we further consider the two cases separately: In the case u i > 0, since u i − u i−1 < 0, we have In the remaining case u i 0, we plainly get Also, we distinguish between the two cases u i > 0 and u i 0. We start with the first case. Since In the latter case u i 0, we have again, In this region, it in turn holds that In this region, we further distinguish between the two cases u i−1 − u i and In the latter case u i−1 − u i > , we have By interchanging of the role of u i (x) and u i−1 (x), we also gain the same estimates as Cases (I), (II), and (IV) in the regions , respectively. Collecting all cases, we finally arrive at that is, This together with the fact that h q−1 1 implies that holds true on \ ( 1 ∪ 2 ). Thereby, we have established (4.5) in the whole region and, the preceding estimates (4.4) and (4.5) yield (4.3). This completes the proof of Lemma 4.1.

Convergence Results for the Truncated Solution
In the subsequent lemma, we state the convergence result on the truncated solution in (4.1). is bounded in L 1 ( T ). According to the Rellich-Kondrachov compactness theorem, there exist a subsequence (which we still denote by u h ) and the limit function (ω ) + ∈ L γ ( T ) depending on , such that

Lemma 4.2 Let
This finishes the proof.
The next lemma allows us to estimate the L γ -norm of (u h ) ± − (u h ) ± .
Proof With r defined above, the Sobolev-Gagliardo-Nirenberg inequality which in turn implies (4.6). The remaining estimate (4.7) now follows from the Hölder inequality and the first estimate (4.6).
We shall prove that ((u h ) ± ) h>0 is a Cauchy sequence in L γ ( T ) for all γ with 1 γ < min r , n+1 n in the subsequent lemma. Proof We prove the statement for ((u h ) + ) h>0 only, as the other case is similarly treated. With γ being 1 γ < min r , n+1 n , we split into three terms: where the meanings of I h, , I h,h , , and I h , are clear from the context. Since holds for (x, t) ∈ × (t i−1 , t i ] and i = 1, . . . , k, the Minkowski inequality yields that We now estimate I 1 since I 2 is proved as well as I 1 . Since . . , k, we obtain from (4.6)-(4.7) in Lemma 4.3 that Hence, Clearly, I h , is estimated similarly as I h, . Inserting the preceding estimates into (4.8) and employing Lemma 4.2, we arrive at Since by γ < r lim →∞ ζ( ) = 0, we finally pass to the limit → ∞ in the above display to get Analogously, the same statement holds for (u h ) − − (u h ) − , which finishes the proof.

Convergence of Approximate Solutions I
Here, we shall present the convergence of the approximate solutions u h andū h .

Lemma 4.5 (Convergence of approximate solutions I)
Let r ∈ 1, np n− p for 1 p < n and r ∈ [1, ∞) for n p < ∞. Then, for any positive T < ∞ and all γ ∈ 1, min{r , n+1 n } , there exist a subsequence (u h ) h>0 , (ū h ) h>0 (still denoted by the same notation) and a limit function u ∈ L γ ( T ) such that whenever γ ∈ 1, min{r , n+1 n } . Hence, passing to a subsequence (also labeled with h) we in turn obtain that u h → u almost everywhere in T , that is the first statement of (4.9).
In order to prove the second statement of (4.9), we employ Lemma 2.4 and the truncated argument, as described in Sect. 4.2. For this, we shall handle the case (ū h ) + only, as the other case is similar. Again, by the Minkowski inequality, we have As described in the proof of Lemma 4.4, we infer that where ζ( ) → 0 as → ∞. Since and, subsequently, taking the limit → ∞ gives Furthermore, (4.10) and (4.13) imply that as h 0. Hence, passing to a further subsequence (also labeled with h), we get (ū h ) + → u + almost everywhere in T in the limit h 0. In an analogous way as done above, the almost convergence (ū h ) − → u − in T is verified. Hence, the second assertion of (4.9) is concluded, finishing the proof.

Convergence of Approximate Solutions II
One of the main tools in this paper is the convergence results summarized in the following Lemmas 4.6, 4.7 and 4.8 .

Lemma 4.6
Let u h ,ū h ,v h , v h , w h , andw h be the approximate solutions of (1.1) defined by (3.3)-(3.4). Then there exist subsequences of them (denoted by the same symbol unless otherwise stated) and a limit function u ∈ L ∞ 0, ∞ ; L q+1 ( ) ∩ W 1, p ( ) such that, the following convergences hold true for any positive T < ∞: (4.14) as h 0.
Proof The proof is based on the energy estimates in Lemmata 3.2, 3.3, and 3.4 and we split the proof into several steps.
Step 1: The proof of (4.14). The Poincaré inequality and estimate (3.6) in Lemma 3.2 imply that (ū h , Dū h ) h>0 is uniformly bounded in L p (0, ∞ ; L p ( ; R n+1 )). Therefore, there exist a (not re-labeled) subsequence and a function u ∈ L p (0, ∞ ; L p ( ; R n+1 )) such that proving the claim (4.14), where, as usual, we use the definition of weak derivatives and by the diagonal argument, this limit function u is equal to that obtained in Lemma 4.5.
Step 3: The proof of (4.17). We further divide this step into two steps: At first, we shall prove the strong convergencew h ≡ |ū h | q−1 2ū h to |u| q−1 2 u in L α ( T ) for all α ∈ [1, 2). Since by the energy estimate (3.6) in Lemma 3.2, (w h ) h>0 is bounded in L 2 ( T ) and, by (4.9) in Lemma 4.5,w h −|u| q−1 2 u → 0 almost everywhere in T , where we use algebraic inequality (2.5) in Lemma 2.3 with α = q+3 2 . This allows us to use (2.8) in Lemma 2.4 to get which proves the first statement of (4.17). Next, the difference betweenw h and w is estimated as and thus, the energy estimate (3.10) in Lemma 3.4 gives which converges to zero as h 0. This together with (4.19) finishes the proof of (4.17).
Step 4: The proof of (4.18). Now we proceed exactly same as in Step 2 and split this stage into two steps.
We again use the energy estimate (3.6) in Lemma 3.2 to see that (v h ) h>0 is bounded in L q+1 q ( T ). From (4.9) in Lemma 4.5, as h 0,v h − |u| q−1 u → 0 almost everywhere in T follows, where we use algebraic inequality (2.5) in Lemma 2.3 with α = q + 1. Hence, with the help of (2.8) in Lemma 2.4, we gain v h → |u| q−1 u strongly in L β ( T ), ∀β ∈ 1, q+1 q and therefore the first statement of (4.18) is actually verified.
To conclude the proof, we shall consider the difference between v h and |u| q−1 u. For this, we now distinguish two cases between q 1 and the opposite case 0 < q < 1. In the first case q 1, since In the latter case 0 < q < 1, in view of (2.5) in Lemma 2.
and, subsequently, exploiting Hölder's inequality twice with pair of exponents 2 q , 2 and using the energy estimates (3.6)-(3.7) in Lemma 3.2, we infer that Passing to the limit h 0 in the preceding estimates (4.20)-(4.21) together with (4.18) 1 , we conclude that the convergence v h → |u| q−1 u in L 1 ( T ), which in turn implies v h → |u| q−1 u almost everywhere in T . This time, (3.8) in Lemma 3.3 tells us that (v h ) h>0 is bounded in L q+1 q ( T ). We therefore apply again (2.8) in Lemma 2.4 to conclude the second statement of (4.18).
Finally, the proof of Lemma 4.6 is complete.
From Lemma 2.4, we easily deduce the following convergence, used later in Sect. 5.
Proof We confine the proof of (4.23) only, as the other case (4.22) is similarly treated. We may suppose the support of ϕ is contained in T for some positive T < ∞. Since by (3.8) in Lemma 3.3 (v h ) h>0 is bounded in L q+1 q ( T ) and by (4.18) Thus, (2.8) as h 0, and therefore, the proof is complete.
In the subsequent lemma, we state the strong convergence of spatial gradient, which is the main output in the paper. Before proceeding, let us give some definitions: For h = 0, we denote the differential quotient of a function v with respect to time by

By this notation, ∂ t v h is written as
Thus, the weak formulation (3.5) is rewritten as In the subsequent lemma, we abbreviateū h ≡ū h (t) and [u] ε ≡ [u] ε (t). Here, [u] ε is defined by (2.1) with h = ε and v 0 = u 0 . In this setting, we deduce the following finite integration by parts formula (see also [4,Lemma 2.10]).

Lemma 4.9
Assume the initial datum u 0 ∈ L q+1 ( ) ∩ W 1, p 0 ( ). Letū h be as in (3.3) and u be as in Lemma 4.6, where u is extended to negative times by u(t) := u 0 for t < 0. Then, the following quantitative estimate holds true: (4.25) where the error terms are given by In particular, for any ε > 0 The proof of this lemma is postponed, and will be presented in Appendix B. Since, by (2.4) in Lemma 2.2, B[·, ·] is non-negative, we cancel the second term in the right-hand side of (4.25) in Lemma 4.9 to get For the first term of (4.26), we use the following convergences : By the integrability of u in Lemma 4.6 and Lemma 2.1-(v), for any positive ε < 1 and, |ū h | q−1ū h → |u| q−1 u almost everywhere on T by (4.9) in Lemma 4.5. These convergences together with Lemma 2.1-(iii) and (2.5) in Lemma 2.3 yield that The second term on the right-hand side of (4.26) is computed as sinceū h (t) = u 0 and [u] ε (t) = u 0 for t ∈ (−h, 0). Combining this with (4.26)-(4.27) and using formula (4.24), we infer that On the other hand, it follows from (3.6) in Lemma 3.2 and Hölder's inequality that Due to the weak convergence (4.14) in Lemma 4.6 At this point, we need to distinguish cases between p 2 and 1 < p < 2. Note that all estimations (4.28)-(4.30) are valid for all 1 < p < ∞.
In the first case p 2, the estimations (4.28)-(4.30) together with (2.7) in Lemma 2.3 imply that Since [Du] ε → Du in L p ( T ) as ε 0 by Lemma 2.1-(ii), passing to the limit as ε 0 in the above display, we get proving the claim in the first case p 2.
We are considering the latter case 1 < p < 2, Hölder's inequality, inequality (2.6) in Lemma 2.3 and estimate (3.6) in Lemma 3.2 yield Taking into account the fact that (4.31) 2 -(4.31) 6 do not require the condition p 2, we infer that 1 2 and letting ε 0 in the above display implies that This completes the proof of Lemma 4.8.

Proof of Theorem 1.1
We are now ready for Theorem 1.1 since we have all prerequisites at hand.
Proof of Theorem 1. 1 We will prove that the limit function u obtained in Lemma 4.6 is a weak solution of (1.1) in the sense of Definition 1. For this we split the proof into several steps. For the sake of readability, we describe the flow chart of whole steps: Step 1 : The condition (D1), first part. Let T be any finite positive number. From convergences (4.14)-(4.16) in Lemma 4.6, energy estimates (3.6)-(3.7) in lemma 3.2 and the lower semi-continuity of norms with respect to the weak and * -weak convergence, we infer that the limit u obtained in Lemma 4.6 belongs to L p (0, ∞ ; W 1, p ( )) ∩ L ∞ (0, ∞ ; L q+1 ( )). We will give the proof of the implication u ∈ C([0, T ] ; L q+1 ( )) in Step 4.
Step 2 : The condition (D2). By using (4.23) in Lemma 4.7 and Lemma 4.8, we pass to the limit as h 0 in the identity (3.5) and obtain which ensures (2.9). Here, we notice that the class of test functions C ∞ 0 ( T ) is a subspace of that in (3.5). The function u also satisfies the following identity:

Lemma 5.1 Let u be obtained in Step 1. Then
We will postpone the proof to Appendix C.
Step 3 : The regularity part. Since by (3.10) in Lemma 3.4 (∂ t w h ) h>0 is bounded in L 2 ( T ), there are a subsequence, still denoted by ∂ t w h , and a limit function ω ∈ L 2 ( T ) such that as h 0. Using convergence (4.22) in Lemma 4.7 and passing to the limit as h 0 in the identity give us that Accordingly, it follows from (5.2) and (5.3) that and thus, the lower semi-continuity with respect to this weak convergence and (3.10) in Lemma 3.4 render that Since the right-hand side is independent of T , sending T ∞ concludes the desired regularity (1.5).
Step 4 : The condition (D1), second part (time continuity of solution). We start with taking t 1 , t 2 ∈ [0, T ] arbitrarily. By Hölder's inequality and (5.4), we infer that We now distinguish the two cases between q 1 and 0 < q < 1. When q 1, we use (2.7) with α = q+3 2 to infer that and thus, this together with (5.5) in turn implies showing our implication u ∈ C([0, T ] ; L q+1 ( )) in the case q 1. In the latter case 0 < q < 1, we apply (2.6) with α = q+3 2 to get which together with (5.5) implies that Using this estimate and Hölder's inequality with the exponent 2 q+1 , 2 1−q (note that 0 < q < 1 ⇐⇒ 2 q+1 > 1), we have Furthermore, the lower semicontinuous with respect to the * -weak convergence (4.15) in Lemma 4.6 and estimate (3.6) in Lemma 3.2 imply that holds true for all i = 1, 2. Collecting the preceding estimates, we infer that, for all which also proves the claim in the case 0 < q < 1. Therefore, the condition (D1) is finally verified.
Step 5 : The condition (D3). In this step, we first show the validity of the initial condition (D3). Since we have proven that u ∈ C [0, T ] ; L q+1 ( ) in the previous step, it is sufficient to show u(0) = u 0 .
Step 6 : The energy structures. Using convergences (4.14) and (4.15) in Lemma 4.6 and the lower semi-continuity of norms with respect to the weak and * -weak convergence, we pass to the limit h 0 in energy estimate (3.6) in Lemma 3.2 to deduce (1.3). Inequality (1.4) similarly follows from convergence (4.16) in Lemma 4.6 and energy boundedness (3.7) in Lemma 3.2.
Therefore, the proof of Theorem 1.1 is finally complete.

Remark 5.2
It is worth remaking that, we have the further initial regularity We shall present the full proof of this identity in Appendix D.
By Young's inequality holds for any w ∈ X. Let (u i ) i∈N be a minimizing sequence satisfying which implies that the sequence (u i ) i∈N is uniformly bounded (with respect to i ∈ N) in W 1, p 0 ( ) ∩ L q+1 ( ). Thus, there exist a subsequence, still denoted by (u i ) i∈N , and a weak limit u ∈ W 1, p Since F is convex, F is lower semicontinuous with respect to the weak convergence in X ≡ W 1, p ( ) ∩ L q+1 ( ) and thus, we arrive at which ensures the existence of the minimizer of F.
Next, we compute the first variation of F. Let u i ∈ X be a minimizer of F. For s ∈ (−1, 1) and every ξ ∈ X, it holds that where the definitions of I 1 -I 3 are obvious in the context. In turn, it sees that while, due to the fundamental theorem of calculus, there holds that and, by Young's inequality, the integrant of the integral on the last line is estimated as Thus, Lebesgue's dominated convergence theorem renders that as s → 0. Similarly, we deduce in the limit s → 0. Combining the three preceding formulae and passing to the limit s → 0 in (A.2), we arrive at for every ξ ∈ X. This finishes the proof of the lemma. (B. 2) The first term on the right-hand side of (B.2) is computed as and the second term on the right-hand side of (B.2) is written as Inserting the preceding formulae into (B.2), we have with the remainder term This joined to the identity We are going to reformulate the remainder term . Using the boundary term B[·, ·], the integrand of the first term of is rewritten as and the second one becomes Furthermore, by the convexity of a function R x → 1 q+1 |x| q+1 ∈ R, we get Merging the preceding formulae and estimate, we therefore have Step 2. Next, we shall prove that lim h 0 (ε) and integrating over T leads to 0 (ε) Performing similarly, we also have Merging these convergences in (B.5), we gain proving the first assertion of (4.26). On the other hand, the integrand of (ε) 2 (h) is estimated as and, the first and second terms on the right-hand side converges as in the limit h 0. Hence, this together with the fact that proving the second claim of (4.9). Therefore, the proof is complete.

Appendix C. Proof of Lemma 5.1
In this appendix, we record the proof of Lemma 5.1.
Proof of Lemma 5. 1 We start with deducing that v h (t) is weakly continuous on t ∈ [0, ∞) in L q+1 q ( ), uniformly on the parameter h. Let t 2 > t 1 0 be arbitrarily taken. Testing the approximating equation (3.5) with ψ(x)1l (t 1 ,t 2 ) (t), where ψ = ψ(x) ∈ C ∞ 0 ( ) and 1l (t 1 ,t 2 ) (t) is the usual Lipschitz approximation of characteristic function of time interval (t 1 , t 2 ). Appealing to Hölder's inequality and (3.6) in Lemma 3.2, we gain for every t 2 > t 1 0. By the density of C ∞ 0 ( ) in L q+1 ( ), Hölder's inequality and (3.6) in Lemma 3.2 again, the above display implies that v h (t) is weakly continuous on t ∈ [0, ∞) in L q+1 q ( ), uniformly on approximation parameter h. For any ε > 0, we define We apply (3.5) with the testing function ζ ε (t)ϕ, where ϕ is a smooth function with support in × [0, T ), to gain Since by definition v h (0) = |u 0 (x)| q−1 u 0 (x) and the weak continuous of v h (t) in L q+1 q ( ), we pass to the limit ε 0 in the above display to have therefore, sending h 0 finally concludes that where we used (4.23) in Lemma 4.7 and Lemma 4.8 again.
This together with the uniform convexity 1 of the Lebesgue space L p (refer to [11,15]) shows that lim i→∞ f i − f L p ( ) = 0, which concludes the proof.
We are now in position to prove (5.6).
The preceding convergences conclude that