The BV solution of the parabolic equation with degeneracy on the boundary

Abstract Consider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.


Introduction and the main results
Yin-Wang [1] first studied the equation u t D div. ˛j ruj p 2 ru/; .x; t / 2 Q T D .0; T /: where is a bounded domain in R N with appropriately smooth boundary, .x/ D dist.x; @ /; p > 1,˛> 0. An obvious character of the equation is that, the diffusion coefficient depends on the distance to the boundary. Since the diffusion coefficient vanishes on the boundary, it seems that there is no heat flux across the boundary. However, Yin-Wang [1] showed that the fact might not coincide with what we image. In fact, the exponent˛, which characterizes the vanishing ratio of the diffusion coefficient near the boundary, does determine the behavior of the heat transfer near the boundary. One may refer to [1] for the details. In our paper, we will consider the following equation u t D div. ˛j ruj p 2 ru/ C N X i D1 @b i .u/ @x i ; .x; t / 2 Q T : The convection term P N iD1 @b i .u/ @x i not only brings the difference on operational skills, but more essentially, it makes the nature of boundary condition change. The equation (2) had been originally studied by the authors in [2,3]. Instead of the whole boundary condition u.x; t / D 0; .x; t / 2 @ .0; T /; only a partial boundary condition u.x; t / D 0; .x; t / 2 † p .0; T /; matching equation (2) is considered. Here, denoting fn i .x/g as the unit inner normal vector of @ , when b i 0 .0/n i .x/ < 0, 8x 2 @ , then † p D @ . But generally, it is just a portion of @ . However, we don't need to pay too much attention to its explicit formula, we only need to remember it is just a subset of @ . Certainly, the initial value is always necessary, u.x; 0/ D u 0 .x/: In [2,3], we said a bounded domain has the integral non-singularity, if the constants˛; p, satisfy Z 2p 2 dx 6 c: We assumed that there are constantsˇ; c such that jb i .s/j 6 cjsj 1Cˇ;ˇb0 i .s/ˇ6 cjsjˇ: If p > 2, we had obtained the existence of the solution of equation (2) with the initial boundary values (4)- (5), and if † p D @ , we also had obtained the stability of the weak solutions. In our paper, we will promote the existence of the solution without the condition (6) but limiting that˛ 1. The most innovation of our paper is that the stability of the weak solutions can be obtained only based on the partial boundary condition (4). Comparing with the case of that † p D @ in [2,3] or † p D ; in [1] (when˛ p 1), how to obtain the stability of the weak solutions only based on the partial boundary condition (4) seems very difficult. Let us give the basic definitions and the main results as following.
Definition 1.1. A function u.x; t / is said to be a local BV solution of equation (2) with the initial value (5), if and for any function ' 2 C 1 0 .Q T /, the following integral equivalence holds The initial value is satisfied in the sense of Here, Q T D f.x; t / 2 Q T W .x/ D dist.x; @ / > g for small enough > 0.   (2), satisfies that when x is near @ , then we say u is a regular solution.
Remark 1.4. If b i .s/ Á 0, we had proved that the solution of equation (1) is regular in [4].
and with the different initial values u.x; 0/ D v.x; 0/ respectively. If b i .s/ is a Lipschitz function, and moreover Here, n > 0 is a nature number, the details of the definition and the properties of the function g n .s/ is in Section 3, in particular, j g n .s/s jÄ c.
Theorem 1.7. Let˛ p 1, and u; v be two local BV solutions of (1) with the initial values u 0 .x/; v 0 .x/ respectively. If u and v are regular, and The most important character of Theorem 1.7 is in that we obtain the stability (15) without any boundary value condition. However, since the solutions considered in the theorem are regular, we can easily obtain the conclusion (15) in a similar way as in [1]. So we omit the details of the proof of the theorem in our paper.
Recently, the author has been interested in the initial-boundary value problem of the following strongly degenerate parabolic equation The stability of the solutions based on a partial boundary condition (4) has been established in [5][6][7] et. al.
Actually, many mathematicians have been interested in the problem, and have obtained many important results of the the stability of the solutions based on a partial boundary condition, one may see the Refs. [8][9][10][11]. Unlike the equation (16), to the best knowledge of the authors, considering the parabolic equation related to the p Laplacian, our paper is the first one to study the stability of the solutions based on a partial boundary condition (4). Of course, whether the condition (12) in Theorem 1.6 and the assumption that u; v are regular in Theorem 1.7 are necessary or not? This is a very interesting problem to be studied in the future. Some other related references, one can refer to Refs. [12][13][14][15][16]. The paper is arranged as following. In Section 1, we have introduced the problem and given the main results of the paper. In Section 2, we prove the existence of the local BV solution. In Section 3, only based on a partial boundary condition, we prove the Theorem 1.6.
Proof. By the maximum principle, there is a constant c only dependent on ku 0 k L 1 . / but independent on ", such that Multiplying (17) by u " and integrating it over Q T , we have For small enough > 0, let D fx 2 W dist.x; @ / > g. Since p > 1, by (22), Differentiating (17) with t , and denoting w D u "t , then rewriting it as @w @t D a ij where a ij D " .jru " j 2 C "/ p 2 2 .ı ij C .p 2/.jru " j 2 C "/ 1 u x i u x j /: Clearly, w satisfies that w.x; t / D 0; .x; t / 2 @ OE0; T /; Denoting that a D .jru " j 2 C "/ p 2 2 ; then minfp 1; 1gaj j 2 Ä a ij i j Ä maxfp 1; 1gaj j 2 : By the maximum principle, due to˛ 1, we have By (23)-(24), we know that u " 2 BV .Q T /, and Then by Kolmogoroff's theorem, there exists a subsequence (still denoted as u " ) of u " , which is strongly convergent to u 2 BV .Q T /. In particular, by the arbitrary of , u " ! u a.e. in Q T .
Hence, by (22) In order to prove u satisfies equation (2), we notice that for any function ' 2 C 1 0 .Q T /, and u " ! u is almost everywhere convergent, so b i .u " / ! b i .u/ is true. Then Now, if we can prove that " for any function ' 2 C 1 0 .Q T /, then u satisfies equation (2).

Remark 2.4. The condition˛
1 is only used to prove that ju "t j Ä c, which implies that R Q T ju "t jdxdt Ä c. Maybe one can prove the later conclusion directly. u 0 2 C 1 0 . / is the simplest condition, but it is not the most general condition. However, we mainly concern with how the degeneracy of diffusion coefficient ˛a ffects the boundary value condition.
where c is independent of n. By a process of limit, we can choose g n . .u v// as the test function, then Z g n . .u v// @.u v/ @t dx C Z ˛.