Electron. J. Differential Equations, Vol. 2020 (2020), No. 74, pp. 1-14

Stability of anisotropic parabolic equations without boundary conditions

Huashui Zhan, Zhaosheng Feng

In this article, we consider the equation

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.

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Huashui Zhan
School of Applied Mathematics
Xiamen University of Technology
Xiamen, Fujian 361024, China
email: 2012111007@xmut.edu.cn
Zhaosheng Feng
School of Mathematical and Statistical Sciences
University of Texas Rio Grande Valley
Edinburg, TX 78539, USA
email: zhaosheng.feng@utrgv.edu

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