The trace of u∈Wloc1,1(Ω)⋂L∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u\in W^{1,1}_{\mathrm{loc}}(\Omega )\bigcap L^{\infty}(\Omega )$\end{document} and its applications

This paper is concerned with the well-posedness problem of a doubly degenerate parabolic equation with variable exponents. By the parabolically regularized method, the existence of local solution is proved. Moreover, the trace of u∈W01,1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u\in W^{1,1}_{0}(\Omega )$\end{document} is generalized to u∈Wloc1,1(Ω)⋂L∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u\in W^{1,1}_{\mathrm{loc}}(\Omega )\bigcap L^{\infty}(\Omega )$\end{document} in a rational way. Then, a partial boundary value condition matching up with the stability theorem is found.


Introduction
Many problems in physics, mechanics, and biology are described by degenerate parabolic equations. For example, the evolutionary equation u t -div a(x)|∇u| p-2 |∇u| + f (x, t, u) = 0 (1) may model the diffusion of a substance in water, soil, or air, heat flow in a material, or diffusion of a population in a habitat. Since the media may not be homogenous, the equation is governed by a diffusion coefficient a(x). When at some points the medium is perfectly insulating, it is natural to assume a(x) = 0 at these points. In fact, certain composite material can block the heat at certain points, or a diffusion of a population may degenerate in some locations due to environmental heterogeneity and barriers [5,7,12]. In this paper, we consider a more complicate evolutionary equation with v(x, t) = 0, (x, t) ∈ ∂ × (0, T), where ⊂ R N is a bounded domain and 0 < T < ∞, Q T = × (0, T), p(x) > 1 is a C 1 ( ) function, both g i (x, t, v) and d (x, t, v) are continuous on Q T × R, a(x) ∈ C( ) is a nonnegative function. Since both a(x) and |v| α(x) may be degenerate, equation (2) is with double degeneracy. Let us take a quick look at some of the progress that has been made. If a(x) = 1, α(x) = α, and p(x) = p are positive constants, equation (2) becomes which is called a polytropic filtration equation with a convection term and a source term. The well-posedness of this equation has been studied widely, one can refer to [6,11,14,24] and the references therein. If a(x) is a nonnegative function satisfying and α(x) ≡ α and p(x) = p are constants, d(x, t, u) = 0, then the stability of weak solutions to equation (2) was studied recently in [25,26]. Also, equation (2) is a simple version of the following equation: which comes from many applied problems such as the electrorheological fluid theory, the system and method for image depositing [1,9,13,18,20]. If 0 < a -≤ a(x, t, v) ≤ a + < ∞, f (x, t, v, ∇v) = b(x, t)|u| σ (x,t)-2 , the existence of local solutions and the blow-up phenomena of equation (7) were studied in [3]. If a(x, t, v) = |v| α +d 0 , d 0 > 0, α ≥ 2, p(x) is continuous with the logarithmic modulus of continuity, f (x, t, v, ∇v) = f (x, t), then the existence and uniqueness of weak solutions were showed in [8]. However, when a(x, t, v) ≥ 0, the uniqueness of weak solution remains open till today. Only when a(x, t, v) = a(x)|v| α(x) , some progress has been made by the author recently. Some details are given in what follows.
However, when α(x) = 0, p(x) = p, g i (x, t, v) = 0, i = 1, 2, . . . , N , and d(x, t, v) = |v| q-1 v with q > p + 1, the solution of equation (2) may blow up in finite time. So, there is only a local weak solution to equation (2). In a word, compared with [22,27,28,30], the main improvements of this paper lie in that the stability of weak solutions is proved when α(x) ≥ 0 but without the assumption α(x) ∈ C 1 0 ( ). Such an improvement is due to the novelty of the classical trace of u ∈ inW 1,1 0 ( ) being generalized to u ∈ W 1,1 loc ( ) L ∞ ( ). The contents are arranged as follows. In the first section, we have given some background. In Sect. 2, the definition of weak solution is introduced and the main results are listed. In Sect. 3, Theorem 2 is proved. The stability theorems are proved in Sect. 4.

The definitions and the main results
To define the weak solution, we give a basic Banach space which can be found in [4]. For every fixed t ∈ [0, T), define the Banach space and denote by V t ( ) its dual space. At the same time, define the Banach space and denote by W (Q T ) its dual space, and define the norm in

Definition 1 A function v(x, t)
is said to be a weak solution of equation (2) with the initial value ∇v ∈ L ∞ 0, T; L p(x) and for any function ϕ ∈ C 1 0 (Q T ), The initial value (3) is satisfied in the sense for any φ(x) ∈ C ∞ 0 ( ).
This definition seems to have nothing to do with the boundary value condition (4) we will specify later. We first give the existence theorem here.

Theorem 2
Suppose that a(x) ∈ C 1 ( ) satisfies (6), g i (x, t, s) and d(x, t, s) are C 1 functions on Q T 0 × R, where σ > 2 and d 0 > 0 are constants, g(x, t) is a C 1 function on then equation (2) with the initial value Certainly, Theorem 2 only tells us the existence of the local solution. If α(x) = 0, a(x) = 1, and p(x) = p is a constant, equation (2) becomes the well-known non-Newtonian fluid equation, when g i (x, t, v) = 0 and |d(x, t, v)| ≤ c|v| p-1 + φ(x, t), φ ∈ L r (Q T ) with r > N+p p , then the existence of global solution was proved in [31], and the same conclusion was obtained in [13] provided that p(x) > 1 is a continuous function. [28,30] recently. If there are not other restrictions on the growth order of d(x, t, v), the weak solutions to equation (2) may blow up, one can refer to [3,10] for details. Let and define for small enough positive constant λ. For simplicity, we call the function ϕ(x), which satisfies (16), a weak characteristic function of . Now, we can generalize the classical trace of v ∈ W 1,1 0 ( ) to that of v ∈ W 1,1 loc ( ) L ∞ ( ) and specify the boundary value condition (4) as follows.

Definition 3 The boundary value condition (4) is true in a general sense of trace if and only if
lim sup where D λ = {x ∈ : ϕ(x) > λ}. Moreover, for any 1 ⊂ ∂ , we define that lim sup where

s) ds
and ϕ(x) ∈ C 1 ( ) be a weak characteristic function of . Then the partial boundary value condition matching up with equation (2) can be imposed as where The main result of this paper is the following theorem. (2) with the initial values u 0 (x), v 0 (x) respectively and with the same partial boundary value condition (20).

Theorem 4 Let u(x, t) and v(x, t) be two solutions of equation
One can see that, since a(x) satisfies (6), ϕ(x) can be chosen as a(x), a(x) k , or e -1 a(x) in Theorem 4. Naturally, the analytical expression ϕ in partial boundary value condition (20) depends on the choice of ϕ. We conjecture that the best partial boundary value condition matching up with equation (2) should be is a weak characteristic function of .

The proof of Theorem 2
Consider the initial-boundary value problem where and for any function ϕ ∈ C 1 0 (Q T ), The initial value (24) is satisfied in the sense as (12).
Then, by a similar method as that in [3], we have the following theorem.
Firstly, we quote the following lemmas.

Lemma 8 Suppose that p(x) ∈ C( ) is local Hölder continuous, and denote that
Then the following facts are true.
This lemma can be found in [9,20] etc. Secondly, we give the details of the proof of Theorem 2.
Proof of Theorem 2 According to Theorem 6, there is a weak solution v ε of the initial boundary value problem (23)- (24), and where T 0 < T * is a given positive constant, c(T 0 ) is a constant that may depend on T 0 .
By this property, one can show that The details are omitted here. Thus, there are functions v(x, t) and Moreover, by the important property (44), it is not difficult to show that for any given function ϕ ∈ C 1 0 (Q T 0 ). Then v is a weak solution of equation (2) with the initial value (3).

Proof of Theorem
The monotonicity of the p(x)-Laplacian operator yields By the definition of ϕ λ (x), there exists By that using the Lebesgue dominated convergence theorem, one has where as η → 0.
Similarly, one has Moreover, for the sixth term on the left hand side of (47), by that and Let (53) minus (54). Firstly, by the definition of ϕ λ , using the Lebesgue dominated convergence theorem, one has Secondly, by the partial boundary value condition (20)-(21), one has Thus, it can be deduced that Thirdly, At last, by that d(x, t, s) is a Lipschitz function, we have After letting η → 0 in (47), let λ → 0. By (49)-(59), we have the well-known Gronwall inequality yields

Conclusion and a simple comment
It is well known that, in order to study the well-posedness problem of a polytropic filtration equation one generally transfers it to the following type: as [6,21], where β = (p -1)(α + p -1) -1 , δ = β p-1 . Then the methods and techniques used in the study of the well-posedness of non-Newtonian fluid equations may be valid. But, since equation (2) contains the nonlinear term |v| α(x) and the variable exponent p(x), to transfer equation (2) to another equation similar to equation (61) is impossible. At the same time, compared with our previous works [27,28,30], the key assumption α(x) ∈ C 0 ( ) in [27,28,30] has been weakened to α(x) ∈ C( ). Moreover, the classical trace of u ∈ W 1,1 0 ( ) is generalized to u ∈ W 1,1 loc ( ) L ∞ (Q T ), and basing on such a generalization, a reasonable partial boundary value condition is found to match up with equation (2). The methods used to prove the stability of the weak solutions also are valid to prove the corresponding stability theorems related to the degenerate parabolic equation appearing in [2,22,27,28,30].
At the end of the paper, we give a simple comment on the definition of the trace. For a linear degenerate elliptic equation [15][16][17] N+1 r,s=1 a rs (x) it is well known that an appropriate partial boundary condition is Here, {n s } is the unit inner normal vector of ∂ and 2 = x ∈ ∂ : a rs n r n s = 0, b ra rs x s n r < 0 , 3 = x ∈ ∂ : a rs n s n r > 0 .
It means that if the matrix ((a rs )) is positive definite, then condition (62) is just the usual Dirichlet boundary condition. Thus, for a classical parabolic equation when the matrix ((a ij )) is positive definite, then we should impose the following initialboundary condition: Naturally, the solutions of equations (60) and (63) are the classical solutions, and conditions (61)(65) are true in the sense of continuity. However, for nonlinear degenerate parabolic equations, the solutions generally are in a weak sense, the boundary value condition cannot be true in the sense of continuity. Moreover, since C ∞ 0 ( ) is dense in a Sobolev space W 1,p 0 ( ), the trace of f (x) ∈ W 1,p 0 ( ) on the boundary ∂ is defined as the limit of a sequence f ε (x) as If the weak solution of a nonlinear equation belongs to a Sobolev space W 1,p 0 ( ), then the Dirichlet boundary value condition is true in the sense of (66). Actually, let BV( ) be the BV function space, i.e., | ∂f ∂x i | is a regular measure, and Then the BV function space is the weakest space such that the trace of u ∈ BV( ) can be defined as (66) (when u = 0 on the boundary ∂ ).
If a weak solution of a nonlinear equation does not belong to a Sobolev space W 1,p 0 ( ), how to impose a suitable boundary condition has been an important and difficult problem for a long time. A typical example is evolutionary p-Laplacian equations of the form where α(x) ∈ C( ), α(x) > 0 in but may be equal to 0 on the boundary ∂ . The authors of [23] classified the boundary ∂ into three parts: the nondegenerate boundary 3 , 3 = x ∈ ∂ : α(x) > 0 , the weakly degenerate boundary 4 = x ∈ ∂ : α(x) = 0, there exists r > 0, such that In [23], the trace of u ∈ B, u(x, t) = 0 on the boundary is defined as where ess sup lim λ→0 f (λ) = inf δ>0 {ess sup{f (λ) : |λ| < δ}} is the super limit.
where n = {n i (x)} is the inner normal vector of ∂ . Along this way, the author of this paper has given another generalization of the trace to the functional space L ∞ (0, T; W 1,p loc ( )) in [29] recently. However, such a generalization of the trace is based on the convection term b i (x)D i u. Once a nonlinear evolutionary equation is without a convection term, for example, if considering the equation v t = div a(x)|v| α(x) |∇v| p(x)-2 ∇v + f (x, t, v), (x, t) ∈ Q T , then the definition of (68) cannot be used. On the other hand, the general trace defined as Definition 3 is valid for equation (70) and any other equations appearing in this paper. So, we think the trace defined as Definition 3 is more natural and novel.