On the evolutionary p-Laplacian equation with a partial boundary value condition

Consider the equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u_{t}} = \operatorname{div} \bigl(d^{\alpha} \vert \nabla u \vert ^{p - 2}\nabla u\bigr) + \frac{\partial b_{i}(u,x,t)}{\partial{x_{i}}},\quad (x,t) \in\Omega \times(0,T), $$\end{document}ut=div(dα|∇u|p−2∇u)+∂bi(u,x,t)∂xi,(x,t)∈Ω×(0,T), where Ω is a bounded domain, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d(x)$\end{document}d(x) is the distance function from the boundary ∂Ω. Since the nonlinearity, the boundary value condition cannot be portrayed by the Fichera function. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha< p-1$\end{document}α<p−1, a partial boundary value condition is portrayed by a new way, the stability of the weak solutions is proved by this partial boundary value condition. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha>p-1$\end{document}α>p−1, the stability of the weak solutions may be proved independent of the boundary value condition.


Introduction and the main results
Benedikt et al. [1] considered the equation u t = div |∇u| p-2 ∇u + q(x)|u| γ -1 u, (x, t) ∈ Q T = × (0, T), (1.1) and showed that the uniqueness of the solution is not true [1]. Here, 0 < γ < 1, is a bounded domain in R N with appropriately smooth boundary, q(x) ≥ 0 and at least there is a x 0 ∈ such that q(x 0 ) > 0. Zhan [2] had shown that the stability of the solutions to the equation is true, where d(x) = dist(x, ∂ ) is distance function, α > 0 is a constant. The result of [2] is in complete antithesis to that of [1]. So, when the well-posedness of the solutions is considered, the degeneracy of the diffusion coefficient d α plays an important role. Yin and Wang [3,4] studied the equation and showed that there is a constant γ > 1 such that, if α < p -1, then Q T |∇u| γ dx dt < ∞. (1.4) Recently, Zhan [5] had generalized the Yin and Wang result to the equation In this paper, we continue to consider a more general equation, and study the well-posedness of the weak solutions. As usual, the initial value is necessary. But, since the coefficient d α is degenerate on the boundary, when α < p -1, though (1.4) is true, and the boundary value condition u(x, t) = 0, (x, t) ∈ ∂ × (0, T), (1.8) can be imposed in the sense of the trace, it may be overdetermined. While α ≥ p -1, it is almost impossible to prove (1.4). How to impose a suitable boundary value condition to match up with Eq. (1.6) becomes very troublesome [4]. Stated succinctly, instead of the Dirichlet boundary value condition (1.8), only a partial boundary value condition, is needed, where p ⊆ ∂ is a relatively open subset. The main difficulty comes from the fact that, since Eq. (1.6) is a nonlinear parabolic equation, p cannot be expressed by the Fichera function (one can refer to Sect. 6 of this paper). In this paper, we will try to depict the geometric characteristic of 1 , and establish the stability of the weak solutions based on the partial boundary value condition (1.9). We denote and Here ϕ 1 ∈ C 1 0 (Q T ), ϕ 2 (x, t) ∈ W 1,p α for any given t, and |ϕ 2 (x, t)| ≤ c for any given x. If the initial value (1.7) is satisfied in the sense of then there is a solution of Eq. (1.6) with the initial value (1.7).
Certainly, we suggest that the conditions in Theorem 1.2 are not the optimal, we only provide a basic result of the existence here. The main aim of this paper is to research the stability of the weak solutions.
(1.15) If u and v are two solutions of Eq. (1.6) with the initial values u 0 (x) and v 0 (x), respectively, then (1.16) Remark 1.4 If α < p -1, we can prove the stability of the weak solutions for the initialboundary value problem (1.6), (1.7), and (1.8) in a standard way [6]. We ask whether the spatial variable x in the nonlinear convection term b i (u, x, t) can bring about the essential change. In particular, when b i (s, x, t) ≡ 0, then only if α ≥ p -1, Yin and Wang [3] had shown that Without the condition (1.15), we can prove a result of the local stability of the weak solutions. This is the following theorem.
(1.17) Theorem 1.5 implies that the uniqueness of the weak solutions is true only if α > 0. When b i (u, x, t) = b i (x)D i u, i.e., the convection term is just linear, Theorem 1.5 had been proved in paper [7]. When b i (u, x, t) = b i (u), Theorem 1.5 had been proved in [8] very recently. For the sake of simplicity, we will not give the details of the proof of Theorem 1.5 in this paper.
Once more, by introducing a new kind of the weak solutions, choosing a suitable test function, we can prove the following theorems. (1.18) If u and v are two solutions of Eq. (1.6) with the initial values u 0 (x) and v 0 (x), respectively, then Last but no the least, we will prove the stability of the solutions based on a partial boundary value condition. (1.20) Here, The paper is arranged as follows. In Sect. 1, we have given the basic definition and introduced the main results. In Sect. 2, we prove the existence of the solution to Eq. (1.6) with initial value (1.7). In Sect. 3, we prove Theorem 1.3. In Sect. 4, we give another kind of the weak solutions. By this new definition, we can prove Theorem 1.6. In Sect. 5, we will prove Theorem 1.7. In Sect. 7, we will give an explanation of the reasonableness of the partial boundary value condition.

The proof of existence
Consider the regularized equation with the initial boundary conditions Here, u 0ε ∈ C ∞ 0 ( ) and u 0ε converges to u 0 in W 1,p 0 ( ).
Proof of Theorem 1.2 Similar to [9], we can easily prove that there exists a weak solution Multiplying (2.1) by u ε and integrating it over Q T , by the fact we are able to deduce that Multiplying (2.5) by u εt , integrating it over Q T , then it yields Notice that Thus, By condition (1.13), By Hölder's inequality and α ≤ p-2 2 , by the inequality, we have Hence, by (2.4), (2.6), (2.11), there exist a function u and a n-dimensional vector Here, if p ≥ 2, r = 2, while 1 < p < 2, 1 < r < Np N-p . In order to prove that u is the solution of Eq. (1.6), for any function ϕ ∈ C 1 0 (Q T ), we have By this note, we have Now, similar to the general evolutionary p-Laplician equation [6], we are able to prove that (the details are omitted here) for any function ϕ ∈ C 1 0 (Q T ). Then α for any given t, and |ϕ 2 (x, t)| ≤ c for any given x, it is clear that ϕ 2 ∈ W 1,p ( ϕ 1 ). By the fact that C ∞ 0 ( ϕ 1 ) is dense in W 1,p ( ϕ 1 ), by a process of limits, we have  We define Since for any given t, ϕ 1 = g n (uv) ∈ W 1,p α , by a process of limit, we can choose d n g n (uv) as the test function, then which goes to zero since that α > p -1. By this fact, |∇d n | = n, x ∈ \ D n , we have which goes to 0 as n → 0.
Once more, since by the Lebesgue dominated convergence theorem, we have Once again, It implies that

Another kind of weak solution
In this section, we introduce another kind of weak solution and prove another stability theorem.

Definition 4.1 If a function u(x, t) satisfies (1.10), and
for ϕ ∈ C 1 0 (Q T ), g(s) is a C 1 function with g(0) = 0, the initial value (1.7) is satisfied in the sense of (1.12), then we say u(x, t) is a weak solution of Eq. (1.6) with the initial value (1.7).
Only if we choose ϕ 1 = g(ϕ), ϕ 2 = 1 in Definition 1.1, one can obtain the existence of the weak solutions in the sense of Definition 4.1.

Theorem 4.2 If b i is a Lipchitz function,
and one of the following conditions is true: is true for the solutions u and v with the initial values u 0 (x) and v 0 (x), respectively.
Proof By a process of limit, we may choose ϕ = χ [τ ,s] g n (d β (uv)) as a test function, where β is a constant to be chosen later. Then Now, let us calculate every term in (4.5). For the first term on the right hand side of (4.5), Clearly, By the fact that |∇d| = 1 is true almost everywhere, α > p -1, we have accordingly, using the Lebesgue dominated convergent theorem and the limit lim n→∞ s × h n (s) = 0, we have which goes to zero as n → ∞.
As for the second term on the right hand side of (5.5), Since for any given ( which goes to zero when n → 0. This is due to 3), using the Lebesgue dominated convergent theorem in (4.10) and using lim n→∞ sh n (s) = 0 again. Meanwhile, also using the dominated convergent theorem, we have which goes to zero provided that one of the conditions (i) and (ii) is true. Here q = p p-1 as usual.

Proof of Theorem 1.7
Proof For a small positive constant λ > 0, define u and v are two weak solutions of Eq. (1.6) with the same partial homogeneous boundary value (1.20) and with the different initial values u 0 (x) and v 0 (x), respectively. According to Definition 4.1, we choose g n (φ(uv)) as the test function. Thus For the terms on the left hand side of (5.2), By the fact that using the Young inequality, we have which goes to 0 as λ → 0, by p -1 > α ≥ p-1 p-2 , implying According to the definition of the trace, by the partial boundary value condition (1.6), Moreover, as in [10], we can prove that In detail, by the Gronwall inequality, we have the symmetric matrix (a rs (x)) has nonnegative characteristic value, to study its wellposedness problem, one only needs to give a partial boundary condition. In detail, let {n s } be the unit inner normal vector of ∂ and denote 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 .
Then, to ensure the well-posedness of Eq. (1.7), Fichera-Oleǐnik theory tells us that the suitable boundary condition is In particular, if the matrix (a rs ) is positive definite, (6.2) is just the usual Dirichlet boundary condition. Considering the classical parabolic equation with the matrix (a ij ) is positive definite, besides the initial condition where − → a = {a i }. According to Fichera-Oleǐnik theory, the optional boundary value condition is u(x, t) = 0, (x, t) ∈ × [0, T), (6.8) with = x ∈ ∂ : a i (x)n i (x) < 0 , (6.9) where n = {n i } is the inner normal vector of . Now, by reviewing the partial boundary value condition (1.24) we have found ⊆ p . (6.10) Though the condition (1.24) may be not the optimal, it is reasonable.

Conclusion
Besides the diffusion coefficient d α being degenerate on the boundary, Eq. (1.6) has a convection term N i=1 ∂b i (u,x,t) ∂x i , which depends on the spatial variable x. Such a characteristic can bring about essential changes on the boundary value condition. A reasonable partial boundary value condition is proposed for the first time, the stability of the weak solutions based on this partial boundary value condition is established. One can see that, if the convection term is independent of the spatial variable x, putting up a reasonable partial boundary condition becomes more difficult. We hope we can solve this problem in our follow-up work.

Funding
The paper is supported by Natural Science Foundation of Fujian province, supported by Science Foundation of Xiamen University of Technology, China.