Dynamics of Partial Differential Equations

Volume 16 (2019)

Number 3

Stability of hyperbolic-parabolic mixed type equations

Pages: 253 – 272

DOI: https://dx.doi.org/10.4310/DPDE.2019.v16.n3.a2

Authors

Huashui Zhan (School of Applied Mathematics, Xiamen University of Technology, Xiamen, Fujian, China)

Zhaosheng Feng (Department of Mathematics, University of Texas Rio Grande Valley, Edinburg, Texas, U.S.A.)

Abstract

In this paper, we are concerned with entropy solution of hyperbolic-parabolic mixed type equations. Although we can define its trace on the boundary $\partial \Omega$, the Dirichlet boundary value condition may be overdetermined. The main feature which distinguishes this paper from other related works lies in the fact that the stability of weak solution is established based on the partial boundary value condition for a more general case where the convection term $\vec{b}$ is dependent on $u$, $x$ and $t$. We also show that in some special cases, the stability of weak solution can be proved without any boundary condition.

Keywords

hyperbolic-parabolic equation, stability, boundary value condition, entropy solution

2010 Mathematics Subject Classification

Primary 35B35, 35K65. Secondary 35L70.

Received 11 February 2019

Published 30 August 2019