The uniqueness of a nonlinear diffusion equation related to the p-Laplacian

Consider a nonlinear diffusion equation related to the p-Laplacian. Different from the usual evolutionary p-Laplacian equation, the equation is degenerate on the boundary due to the fact that the diffusion coefficient is dependent on the distance function. Not only the existence of the weak solution is established, but also the uniqueness of the weak solution is proved.


Introduction and the main results
Recently, we noticed that Benedikt et al. [1] had studied the equation u t = div |∇u| p-2 ∇u + q(x)u γ , (x, t) ∈ Q T × (0, T), (1.1) and shown that the uniqueness of the solutions of equation (1.1) is not true. Here, is an open bounded domain with a smooth boundary, 0 < γ < 1, p > 1, q(x) ∈ C 1 ( ), q(x) ≥ 0 and there exists at least a point x 0 ∈ , q(x 0 ) > 0. This comes more or less as a surprise. In general, we may think that the source time q(x)u γ only affects the existence of the weak solutions. At the same time, in [2], we have considered the following equation: and we have shown that the uniqueness of the weak solution is true when f (u, ·, ·) is a Lipschitz function, here α > 0, ρ(x) = dist(x, ∂ ) is the distance function from the boundary. Certainly, since 0 < γ < 1 in equation (1.1), f (u, x, t) = q(x)u γ is not a Lipschitz function about the variable u. Consequently, the results in [1] and [2] are compatible. If α = 0, there are a great deal of papers devoted to equations (1.2), many of them are important and interesting. But it is impossible to list all these papers, and we only list a few of them [3][4][5][6][7] here.
In this paper, we assume that q(x) ∈ C 1 ( ). We will consider a nonlinear convectiondiffusion equation related to the p-Laplacian, where 0 < γ < 1. The initial value condition only a partial boundary condition, should be imposed generally, where p ⊆ ∂ is a relatively open subset in ∂ . One can refer to our previous work [2,8].
Since equation (1.3) is a nonlinear equation, it is difficult to depict p as the linear degenerate parabolic equation by the Fichera function. The main aim of this paper is to prove the uniqueness of the solutions without any boundary value condition.
In the first place, since we had known the interesting result of [1] (i.e. the nonuniqueness of the weak solution of equation (1.1)), we should clarify why the uniqueness of the weak solutions of equation (1.3) can be obtained. Let us introduce some basic functional spaces. For every fixed t ∈ [0, T), we define the Banach space and we denote by V t ( ) its dual. Also, we denote the Banach space and we denote by W (Q T ) its dual. According to Antontsev-Shmarev [9], we know Basing on these functional spaces, we can give the definition of the weak solution.
and, for any function ϕ ∈ L ∞ (0, T; W The initial value is satisfied in the sense that The most important of Definition 1.1 lies in u t ∈ W (Q T ). Once the weak solution comes with this property, then we have Lemma 3.1 below, and just by this lemma, we can prove the uniqueness. By comparing the analysis in [1], we know the weak solution defined in [1] does not have this property.
Second, we introduce the existence result. Last but not least we will prove the following local stability.
In particular, for any small enough constant λ > 0, Here, λ = {x ∈ : dist(x, ∂ ) > λ}, by the arbitrariness of λ, we have the uniqueness of the solution. This conclusion implies that the degeneracy of the diffusion coefficient can take place of the usual boundary value condition.
We would like to suggest that, if ρ α is substituted by a nonnegative diffusion coefficient a similar conclusion to Theorem 1.3 is still true. For some special cases, one can see our recent work [10]. Actually, we had used some ideas of [10] to prove Theorem 1.3. This paper is arranged as follows. In Section 1, we give the basic definition and introduce the main results. In Section 2, we prove the existence of the solution to equation (1.1) with initial value (1.4). In Section 3, we prove Theorem 1.3 and obtain the uniqueness of the solution.
We now consider the following regularized problem: since 0 < γ < 1, it is well known that the above problem has an unique classical solution [12,13]. By the maximum principle, there is a constant c only dependent on u 0 L ∞ ( ) but independent on ε, such that Multiplying (2.1) by u ε and integrating it over Q T , we get It is also easy to show that by Young inequality, we can show that and so ∇(ϕu ε ) p,Q T ≤ c.
(2.11) By Lemma 2.1, ϕu ε is relatively compact in L s (Q T ) with s ∈ (1, ∞). Then ϕu ε → ϕu a.e. in Q T . In particular, due to the arbitrariness of λ, u ε → u a.e. in Q T . Hence, by (2.4), (2.7), there exists a function u and an n-dimensional vector function In order to prove that u satisfies equation (1.3), we notice that, for any function ϕ ∈ C ∞ 0 (Q T ), and u ε → u is almost everywhere convergent, for any function ϕ 1 ∈ C ∞ 0 (Q T ), then u satisfies equation (1.3). In what follows, we will use a similar method to that in [14] to prove (2.14).
Proof of Theorem 1.3 Let u, v be two solutions of equation (1.3) with the initial values u 0 (x), v 0 (x), respectively. Denote λ = {x ∈ : dist(x, ∂ ) > λ}, let the constant β ≥ max{ p-α p-1 , 2, αp} and We may choose χ [τ ,s] (u ε -v ε )ξ λ as a test function, where u ε and v ε are the mollified function of the solutions u and v, respectively. Then where Q τ s = × (τ , s). For any give λ > 0, since ∇u ∈ L p ( λ ), ∇v ∈ L p ( λ ), according to the definition of the mollified function u ε and v ε , we have (3.5) Let us analyze every term in (3.3). For a start, we deal with the first term on the right hand side of (3.3). Since on λ , by the weak convergency of (3.5) By (3.4)-(3.5), using the Lebesgue dominated convergence theorem, Here, we have used the fact that |∇ρ| = 1 is true almost everywhere. Now, by β ≥ p-α p-1 , we have Now we deal the second term on the right hand side of (3.3). By the Lebesgue dominated convergence theorem and the Hölder inequality (3.12)