STABILITY OF ANISOTROPIC PARABOLIC EQUATIONS WITHOUT BOUNDARY CONDITIONS

. In this article, we consider the equation u t = N (cid:88) i =1 (cid:0) a i ( x ) | u x i | p i ( x ) − 2 u x i (cid:1) x i , with a i ( x ) ,p i ( x ) ∈ C 1 (Ω) and p i ( x ) > 1. Where a i ( x ) = 0 if x ∈ ∂ Ω, and a i ( x ) > 0 if x ∈ Ω, without any boundary conditions. We propose an analytical method for studying the stability of weak solutions. We also study the uniqueness of a weak solution, and establish its stability under certain conditions.

Throughout this paper, we assume that p 0 > 1.Before stating our main results, let us recall two definitions.
and for then we call u(x, t) a weak solution of equation (1.2) with the initial condition (1.3) in the sense of Here, L pi(x) (a i , Ω) is the weighted variable exponent Lebesgue space.One can refer to [11] for the definition of such a space and the corresponding Hölder inequality.
Recall that the characteristic function χ of Ω is defined by Apparently, the weak characteristic function is not unique for a bounded domain Ω.For examples, the distance function d(x) = dist(x, ∂Ω) and the diffusion function a i (x) in (1.6) both are the weak characteristic functions.Based on Definition 1.2, we propose a new analytical method, currently called the weak characteristic function method, to study the stability of weak solutions to the nonlinear degenerate parabolic equations independent of the boundary condition.
Theorem 1.3.Let a i (x) ∈ C 1 (Ω) satisfy (1.6), and u(x, t) and v(x, t) be two solutions of equation (1.2) with the initial values u 0 (x) and v 0 (x) respectively.If for sufficiently large n, there are a weak characteristic function χ(x) of Ω and a constant c such that then where p + i = max x∈Ω p i (x) and Ω n = {x ∈ Ω : χ(x) > 1/n}.Theorem 1.4.Let a i (x) ∈ C 1 (Ω) satisfy (1.6), and u(x, t) and v(x, t) be two weak solutions of (1.2) with the initial values u 0 (x) and v 0 (x) respectively, If there exists a weak characteristic function χ such that then the stability (1.12) is true.
Theorem 1.5.Let a i (x) ∈ C 1 (Ω) satisfy (1.6), and u(x, t) and v(x, t) be two solutions of (1.2) with the different initial values u 0 (x) and v 0 (x) respectively, but without any boundary condition.If there exist a weak characteristic function χ(x) and a constant c such that then This inequality implies that the uniqueness of weak solution is always true provided that a i (x) satisfies conditions (1.5) and (1.6).Note that by choosing various characteristic functions χ(x), one may obtain different results.For example, choosing then we obtain From Theorem 1.3 we obtain the following result.
Corollary 1.6.Let a i (x) ∈ C 1 (Ω) satisfy (1.6), and u(x, t) and v(x, t) be two solutions of equation (1.2) with the initial values u 0 (x) and v 0 (x) respectively.If for the sufficiently large n, it holds then the stability (1.12) is true.
Similarly, since by Theorem 1.4, we have the following result.
Corollary 1.7.Let a i (x) ∈ C 1 (Ω) satisfy (1.6), and u(x, t) and v(x, t) be two weak solutions of equation (1.2) with the initial values u 0 (x) and v 0 (x) respectively, If there exists a characteristic function χ(x) such that then the stability (1.12) is true.
If a i (x) ≡ a(x), then condition (1.14) holds, i.e. equation (1.2) reduces to From Theorem 1.5, we have the following result.
Corollary 1.8.Let a(x) ∈ C 1 (Ω) satisfy (1.5) and u(x, t) and v(x, t) be two solutions of equation (1.18) with the differential initial values u 0 (x) and v 0 (x) respectively. Then If a i (x) ≡ a(x) and p i (x) ≡ p, then condition (1.16) is equivalent to condition (1.17), which is also equivalent to In this case, equation (1.2) reduces to ) is true, then the stability (1.12) is true without any boundary condition.As we can see, equation (1.20) is different from the evolutionary p-Laplacian equation: It is notable that if we choose appropriate weak characteristic functions, we can obtain nice results on the stability.One can see that the weak characteristic function method can also be generalized to study the stability of weak solutions to a more general degenerate parabolic equation as well as the evolutionary p-Laplacian equations.
The remainder of this paper is structured as follows.In Sections 2-4, we prove Theorems 1.3-1.5 respectively, by means of the proposed weak characteristic function method.In Section 5, we extend this method to study the stability of solutions of the evolutionary p-Laplacian equation (1.21).
For n > 0, let Obviously, h n (s) ∈ C(R), and Let u(x, t) and v(x, t) be two weak solutions of equation (1.2) with the initial values u 0 (x) and v 0 (x) respectively, but without any boundary condition.Let χ(x) be a weak characteristic function of Ω.We define where Ω n = {x ∈ Ω : χ(x) > 1 n }.By a process of limit, we choose and take χ [τ,s] φ n g n (u − v) as the test function.Here, χ [τ,s] is the characteristic function of [τ, s) ⊆ [0, T ).Then we have (2.4) In the third term of the left-hand side of (2.4), we note that For the first term of the left hand side of (2.4), in view of u t ∈ L 2 (Q T ), it follows the Lebesgue dominated convergence theorem that where q i (x) = pi(x) pi(x)−1 and q + i = max x∈Ω q i (x).Therefore, (2.7) Let η → 0 in (2.4).Then we have Because of the arbitrariness of τ , we obtain

Proof of Theorem 1.4
Making a minor modification, we can generalize Definition 1.1 to the following version.
Let u(x, t) and v(x, t) be two weak solutions of (1.2) with the initial values u 0 (x) and v 0 (x) respectively, and χ be a weak characteristic function.We choose g n (χ(u − v)) as the test function in Definition 3.1.Then we have Let us evaluate each term in the left hand side of (3.2).For the first two terms, we find that and where p i1 = p + i or p − i based on (iii) of Lemma 2.1, and similar for q i1 .If {x ∈ Ω : |u − v| = 0} has zero measure, since we derive that and lim In view of (2.2) and condition (1.13), it follows the Lebesgue dominated convergence theorem that We now letting η → 0 in (3.2), we have By Gronwall's inequality, we obtain 4. Proof of Theorem 1.5 Let u(x, t) and v(x, t) be two weak solutions of equation (1.2) with the initial values u 0 (x) and v 0 (x) respectively.Then we have where χ [τ,s] is the characteristic function on [τ, s] and χ(x) is a weak characteristic function of Ω. Denote Clearly, it has Evaluating the second term on the right hand side of (4.2) yields and In view of (4.2)-(4.7),letting λ → 0 in (4.1) leads to where q < 1.By (4.8), it is easy to see that Due to the arbitrariness of τ , we obtain

Stability of p-Laplacian equation
In the preceding two sections, we use the weak characteristic function method to prove Theorems 1.3-1.5.In this section, we consider equation (1.21) with the initial value condition (1.3), but without any boundary condition.We apply the proposed weak characteristic function method to prove the stability of solutions of equation (1.21).
Proposition 5.1.Let a(x) ∈ C 1 (Ω) satisfy (1.5), and u(x, t) and v(x, t) be two weak solutions of equation (1.21) with the initial values u 0 (x) and v 0 (x) respectively.When p > 1, for the sufficiently large n, it holds where c is a constant.Then the stability (1.12) is true.
Proof.Let χ(x) = [a(x)] N .We can choose φ n g n (u − v) as the test function, then (5.2) Clearly, we see that By a straightforward computations, we derive that which approaches 0 as n → ∞.Hence, by (5.2)-(5.4), the desired result is obtained.
So we can choose an α such that lim Consequently, using (5.9)-(5.12),we arrive at the desire result.Therefore, we can obtain the following proposition which is identical to the corresponding result of [18].
Zhaosheng Feng School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA Email address: zhaosheng.feng@utrgv.edu