The partial boundary value conditions of nonlinear degenerate parabolic equation

The stability of the solutions to a parabolic equation ∂u∂t=ΔA(u)+∑i=1Nbi(x,t)Diu−c(x,t)u−f(x,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\partial u}{\partial t} = \Delta A(u) +\sum_{i=1}^{N}b_{i}(x,t)D_{i}u-c(x,t)u-f(x,t) $$\end{document} with homogeneous boundary condition is considered. Since the set {s:A′(s)=a(s)=0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{s: A'(s)=a(s)=0\}$\end{document} may have an interior point, the equation is with strong degeneracy and the Dirichlet boundary value condition is overdetermined generally. How to find a partial boundary value condition to match up with the equation is studied in this paper. By choosing a suitable test function, the stability of entropy solutions is obtained by Kruzkov bi-variables method.


The boundary condition
We consider the equation and assume that where ⊂ R N is an appropriately smooth bounded domain, D i = ∂ ∂x i , b i (x, t) ∈ C 1 (Q T ), c(x, t), f (x, t) ∈ C(Q T ). Equation (1) has a widely applied background, for example, the reaction diffusion problem [10] and the spread of an epidemic disease in heterogeneous environments.
For the Cauchy problem of equation (1), the paper [2] by Vol'pert and Hudjaev was the first one to study its solvability. Since then there have been many papers to study its well-posedness ceaselessly, one can refer to book [18] and references [1-6, 8, 9, 11, 13-20], and [25,26].
If we want to consider the initial boundary value problem of equation (1), the initial value condition is always necessary. But the Dirichlet homogeneous boundary value condition may be overdetermined generally. In [21,22,24], a version of equation (1) ∂u was studied. Instead of it, a partial boundary value is enough, where 1 ⊂ ∂ is a relative open subset. One can refer to [21,22,24] for details, in which the equation 1 ⊂ ∂ was depicted out in some special ways. Such a fact was found firstly in [19], in which the non-Newtonian fluid equation was considered. Here p > 1, However, in [21] and [22], because there are two parameters including in 1 , the expression 1 seems very complicated and hard to be verified, and the stability of entropy solutions is proved under the assumptions Here d(x) = dist(x, ∂ ) and λ = {x ∈ , d(x) < λ}, λ is small enough, while [24] considered the case of being unbounded and satisfying some harsh terms. Moreover, all partial boundary value conditions appearing in [19,21,22] and [24] are with the form as (6). The dedications of this paper lie in that, for any given bounded domain , due to the fact that the coefficient b i (x, t) depends on the time variable t, we find that, unlike (6), the partial boundary value condition matching up with equation (1) must be of the following form: where p is just a submanifold of ∂ × (0, T) and it cannot be a cylinder as 1 × (0, T). By choosing some technical test functions, the stability of entropy solutions is established by Kruzkov's bi-variables method.

The definition and the main results
For small η > 0, let Obviously, h η (s) ∈ C(R) and Let Define that u ∈ BV(Q T ) if and only if u ∈ L 1 loc (Q T ) and where h = (h 1 , h 2 , . . . , h N , h N+1 ). This is equivalent to that the generalized derivatives of every function in BV( ) are regular measures on . Under the norm BV( ) is a Banach space. A basic property of BV function is that (see [17,18]): if f ∈ BV(Q T ), then there exists a sequence {f n } ⊂ C ∞ (Q T ) such that So, we can define the trace of the functions in a BV space as in a Sobolev space i.e. the trace of f (x) ∈ BV(Q T ) on the boundary ∂ is defined as the limit of a sequence f n (x) as follows: Then it is well known that the BV function space is the weakest space such that the trace of u ∈ BV(Q T ) can be defined as (11) and the integration by parts can be used. Also, one can refer to [7] for the definition of the trace of u ∈ BV(Q T ) on the boundary value in another way.

Definition 1
A function u is said to be the entropy solution of equation (1) with the initial value condition (3) and with the boundary value condition (9) if 1. u satisfies 2. For any ϕ ∈ C 2 0 (Q T ), ϕ ≥ 0, for any k ∈ R, for any small η > 0, u satisfies 3. The partial homogeneous boundary value condition (9) is true in the sense of trace.

The initial value condition (3) is true in the sense that
If and f (x, t) are bounded functions, the existence of the entropy solution in the sense of Definition 1 can be proved by a similar way as that in [21,26], we omit the details here.
In this paper, we study the stability of the entropy solutions of equation (1) without condition (8). In order to display the method used in our paper, the unite n-dimensional cube is considered firstly. By choosing special test functions, we can prove the following theorem.
, respectively, and with the same partial boundary value condition Then is true, then Secondly, we generalize Theorem 2 to a general bounded domain .
The main result of this paper is the following stability theorem. (1) with the initial values u 0 (x) ∈ L ∞ ( ) and v 0 (x) ∈ L ∞ ( ), respectively, and with the same partial boundary value condition

Theorem 3 Suppose that is a bounded smooth domain in R N , and when x is near to the boundary ∂ , the distance function ρ(x) is a C 2 function, A(s) is a Lipschitz function, and a(0) = 0. Let u(x, t) and v(x, t) be the solutions of equation
Then we have Here, where n = {n i } (i = 1, 2, . . . , N) is the outer normal vector of .
We give a simple comment. For a linear degenerate parabolic equation where which implies equation (21) is only degenerate on the boundary ∂ , according to the Fichera-Oleinik theory [12], the optimal boundary value condition matching up with equation (21) is For the nonlinear degenerate parabolic equation (1), the most important characteristic lies in that the set {s ∈ R : a(s) = 0} may have interior points, and so it is a strongly degenerate parabolic equation. In addition, when the Dirichlet boundary value condition (4) is imposed, a(0) = 0 exactly implies that equation (1) is degenerate on the boundary On the other hand, when a partial boundary value condition (9) is imposed, we only know that (1) is degenerate on the boundary p , while on T \ p , whether equation (1) is degenerate or not is uncertain. To the best knowledge of the author, this is the first paper to study the stability of entropy solutions to equation (1) when the partial boundary value condition is imposed on a submanifold p ⊂ T .

An important inequality
In this section, we use the Kruzkov bi-variables method to deduce an important inequality. Such a method was used in [18,21,26] and many other references. We begin with some basic denotations. For u ∈ BV(Q T ), we denote by that u is the set of all jump points, ν is the normal of u at X = (x, t), u + (X) and u -(X) are the approximate limits of u at X ∈ u with respect to (ν, Y -X) > 0 and (ν, Y -X) < 0, respectively. For a continuous function f (u), the composite mean value of f is defined as When f (s) ∈ C 1 (R), u ∈ BV (Q T ), by [17,18], we know f (u) ∈ BV(Q T ) and Just as that in [23,26], we have the following lemma.

Lemma 4
Let u be an entropy solution of (1). Then where I(α, β) denotes the closed interval with endpoints α and β, and (23)  From Definition 1, for any ϕ ∈ C 2 0 (Q T ), we have Let ω h be the mollifier which is defined as Now, we choose ϕ = ψ in (24) and (25), then we have Since Meanwhile, we have where By that and using Lemma 4, we obtain the facts and Then we have as η → 0.
Once more, Also, we clearly have Letting η → 0, h → 0 in (26) and combining (27)-(34), we get This is the most important inequality to prove Theorem 2 and Theorem 3.

Proof of Theorem 2
The proof of Theorem 2 Let For small enough λ, we set Let 0 ≤ η(t) ∈ C 2 0 (t) and Then Then (36) yields We now substitute these formulas into (35), if b i (x, t) ≥ 0, by (38), we have Accordingly, we have According to the definition of the trace of BV functions (see [7]), when We also have (40).
Thus, by the Kruzkov bi-variables method, we have proved Theorem 2.

Proof of Theorem 3
Proof of Theorem 3 Let u(x, t) and v(x, t) be two entropy solutions of equation (1) with the initial values Recalling (35), for any φ(x, t) ∈ C 2 0 (Q T ), we have Since when x is near to the boundary ∂ , the distance function ρ(x) is a C 2 function, we can define where 0 ≤ η(t) ∈ C 2 0 (t) and Then we have and where 0 ≤ ρ(x) < πλ 2 for small λ.
we have Notice that We denote where n = {n i } (i = 1, 2, . . . , N ) is the outer normal vector of . At the same time, we have Processing in an analogous manner as we did in the discussion of (40), letting λ → 0 in (42), we arrive at the desired result.
Thus, we have proved Theorem 3 by the Kruzkov bi-variables method. One can see that the partial boundary value condition is imposed on a submanifold p ⊂ ∂ × (0, T). Such a conclusion reflects how the time variable t affects the well-posedness problem of a degenerate parabolic equation. By the way, the assumption that, when x is near to the boundary ∂ , the distance function ρ(x) is a C 2 function can be weakened as follows: there is a subdomain λ = {x ∈ : ρ(x) < λ}, ρ(x) is an almost everywhere C 2 function on λ , and λ | ρ| dx ≤ c.