On a doubly degenerate parabolic equation with a nonlinear damping term

Consider a double degenerate parabolic equation arising from the electrorheological fluids theory and many other diffusion problems. Let vε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v_{\varepsilon }$\end{document} be the viscous solution of the equation. By showing that |∇vε|∈L∞(0,T;Llocp(x)(Ω))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|\nabla v_{\varepsilon }|\in L^{\infty }(0,T; L_{\mathrm{loc}}^{p(x)}(\Omega ))$\end{document} and ∇vε→∇v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\nabla v_{\varepsilon }\rightarrow \nabla v$\end{document} almost everywhere, the existence of weak solutions is proved by the viscous solution method. By imposing some restriction on the nonlinear damping terms, the stability of weak solutions is established. The innovation lies in that the homogeneous boundary value condition is substituted by the condition a(x)|x∈∂Ω=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a(x)| _{x\in \partial \Omega }=0$\end{document}, where a(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a(x)$\end{document} is the diffusion coefficient. The difficulties come from the nonlinearity of |∇v|p(x)−2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vert {\nabla v} \vert ^{p(x)-2}$\end{document} as well as the nonlinearity of |v|α(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|v|^{\alpha (x)}$\end{document}.

Let a(x) satisfy a(x) = 0, x ∈ ∂ , a(x) > 0, x ∈ . (1.10) Then equation (1.1) is degenerate on the boundary ∂ . If α(x) = 0, p(x) = p is a constant and f (x, t, v, |∇v|) = 0, on the stability of weak solutions, that the degeneracy of a(x)| x∈∂ may take place of the usual boundary value condition (1.3) was revealed in [20,21]. Moreover, whether similar results have been obtained in [27] and [25] respectively. For the other related papers, one can refer to [19,23,24] ∂x i is replaced by a nonlinear damping term f (x, t, v, |∇v|). Considering all these factors, compared the damping term f (x, t, v, |∇v|) with the degeneracy of a(x)| x∈∂ , the latter plays a leading role when the uniqueness problem is considered. Maybe such a conclusion can be explained by the fact that equation (1.1) represents the model that the diffusion process is more dominant than the damping phenomena. For example, for an epidemic model of diseases, it is impossible to know in advance that v = 0 on the boundary ∂ . Thus, imposing the boundary value condition (1.3) seems unreasonable, while the condition a(x)| x∈∂ = 0 can be explained as some anthropogenic interferences are made to control the epidemic across the border ∂ . In accord with this fact, in theory, we conjecture that under the condition a(x)| x∈∂ = 0, one can deduce that v = 0 on the boundary ∂ . This conjecture was partially proved in [22] several years ago, and we are not ready to discuss this conjecture in this paper for the time being.
The main aim of this paper is to establish the well-posedness theory for equation (1.1). To accomplish this aim, the nonlinearity of |v| α(x) and the nonlinearity of the damping term f (x, t, v, |∇v|) are the main difficulties to overcome. The extinction, the positivity, the large time behavior of the solutions and v = 0 on the boundary ∂ , all these important contents remain to be studied in the future.
Let us give the definition of weak solution.
and for any function ϕ ∈ C 1 0 (Q T ), for any φ(x) ∈ C ∞ 0 ( ), then we say that v(x, t) is a weak solution of equation (1.1) with initial value (1.2).
Here, the basic Banach space W(Q T ) and its dual space W (Q T ) are defined by Antontsev and Shmarev in [2]. In addition, let and set q(x) = p(x) p(x)-1 as usual. The main results in this paper are the following theorems.

Theorem 1.4 Let u(x, t) and v(x, t) be two solutions of equation
Moreover, since the diffusion coefficient a(x) satisfies (1.10), we can obtain a stability theorem without the boundary value condition (1.18). (1.2) with the initial values u 0 (x) and v 0 (x) respectively. If α(x) ∈ C 1 0 ( ), a(x) satisfies 24) and the nonlinear damping term satisfies (1. 16) and (1.19), then the stability of weak solutions is true in the sense of (1.21).

Theorem 1.5 Let u(x, t) and v(x, t) be two solutions of equation
Here and in what follows, λ > 0 is a small enough constant, and we define λ = {x ∈ : a(x) > λ}.
Compared with Theorem 1.3, there is not boundary value condition (1.18) in Theorem 1.4. Instead, condition (1.24) is imposed. Comparing with other related works [2,3], the most distinctive assumption in this paper is that α(x) ∈ C 1 0 ( ). Since is always true, and in particular inf x∈ α(x) = 0, but max x∈ α(x) can be larger than p + p + -1 . This fact implies that when α(x) ∈ C 1 0 ( ), u α(x) is beyond the restriction (1.9). So, Theorem 1.3 and Theorem 1.4 have some essential improvements from the works [2,3]. In the next research, we will try to do some work when α(x) is not limited to C 1 0 ( ). By the way, from [4,5] [33] and [31], in order to obtain the well-posedness of weak solutions to equation (1.1), the damping term f (x, t, u, ∇u) must satisfy some restrictions, for example, condition (1.19) and condition (1.20) in our paper. A similar condition was first introduced by Karlsen and Ohlberger in their paper [10], in which the uniqueness of weak solutions to the equation is proved. Although, as one of the reviewers pointed out, condition (1.19) one can refer to [3,7,18]for details.
This lemma can be found in [17].
Proof of Theorem 1.2 At first, let us multiply (2.1) by v ε . Since f (x, t, v, |∇v|) ≤ 0 when v < 0 and satisfies (1.16), by the Young inequality, we have: (2.9) According to condition (1. 16) or (1.17), by the Young inequality, we easily deduce (2.10) as well as for any s ∈ (1, ∞). By (2.13)-(2.14), ϕv p(x) +1 ε → ϕv 1 a.e. in Q T . Due to the arbitrariness of λ, we know v Now, we want to show the local integral of ∇v. For any φ( (2.16) We have the following facts: which goes to zero as ε → 0. By (2.17)-(2.18), we can deduce that which implies that and we have In the end, the initial value is true in the sense of (1.14) can be shown as that of [1]. Thus, v is a weak solution of equation (1.1) in the sense of Definition 1.1.

The global stability
For small η > 0, we define g η (x) to be an odd function, when s ≥ 0, g η (x) has the form Proceeding as in [28], we can prove the following lemma, we omit the details here.
The following lemma is the basic characteristics of the variable exponent Sobolev spaces [6,12,32]. p(x) + 1 q(x) = 1. Then, for any u ∈ L p(x) ( ) and v ∈ L q(x) ( ), we have If u L p(x) ( ) < 1, then u (1.1) with the initial values u 0 (x) and v 0 (x) respectively and with the same homogeneous boundary value condition (1.18). If α(x) ∈ C 1 0 ( ), the nonlinear damping term satisfies

2)
and one of the following conditions is true: Proof We only give the proof of case (A). Case (B) can be proved in a similar way, we omit the details. Since u(x, t) and v(, t) satisfy the same homogeneous boundary value condition (1.18), we can choose g η (uv) as the test function. Then There are two facts much in evidence in (3.5). One is that, by Lemma 3.1, we have Another one is that, by the monotonicity of the operator |∇u| r-2 ∇u, we have Let us discuss the other terms in (3.5). In the first place, α(x) ∈ C 1 0 ( ), we set α = {x ∈ : α(x) > 0} and define Since (2.23) yields |∇u| p(x) , |∇v| p(x) ∈ L 1 loc (Q T ), using the fact lim η→0 g η (s)s = 0 and the Lebesgue dominated convergence theorem, we have and similarly According to (3.8)-(3.9), we can obtain where p + and q + follow from (iii) of Lemma 3.2.
Proof of Theorem 1.3 If the nonlinear damping term satisfies (1.19) we easily show that there is a constant l > 1 such that Proceeding as in the proof of Theorem 3.3, we have the conclusion. If the nonlinear damping term satisfies (1.20), we can prove the conclusion in a similar way, and we do not repeat the details here.
Proof Since u(x, t) and v(, t) satisfy the same homogeneous boundary value condition (1.18), we can choose (uv) as the test function. Then

The global stability if a(x) 1-p(x) dx < ∞
Recalling that, by a weak characteristic function χ(x) of , χ(x) ∈ C( ) and we can set another weak characteristic function as In this section, we explore the stability of weak solutions by the weak characteristic function method [29,30].