Stability of anisotropic parabolic equations without boundary conditions

Abstract

In this article, we consider the equation ut = N∑i=1 (αi(x)|uxi |pi(x)-2 uxi)xi, with αi(x), pi(x) ∈ C1 (Ω̅) and pi(x) > 1. Where αi(x) = 0 if x ∈ ∂Ω, and αi(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.