Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 74, pp. 1-14

Title: Stability of anisotropic parabolic equations without boundary conditions
       
Authors: Huashui Zhan  (Xiamen Univ. of Technology, Fujian, China)
         Zhaosheng Feng (Univ. of Texas Rio Grande Valley, Edinburg, TX, USA)

Abstract:
 In this article, we consider the equation
 $$
 u_t= \sum_{i=1}^N \big(a_i(x)|u_{x_i}|^{p_i(x)-2}u_{x_i}\big)_{x_i},
 $$
 with $a_i(x), p_i(x)\in C^1(\overline{\Omega})$ and $p_i(x)>1$.
 Where $a_i(x)=0$ if $x\in\partial \Omega$, and $a_i(x)>0$ if $x\in \Omega$, 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.
 
Submitted December 9, 2019. Published July 15, 2020.
Math Subject Classifications: 35K15, 35B35, 35K55.
Key Words: Parabolic equation; boundary condition; stability; Holder inequality.