Stability of parabolic equations in non-cylindrical domains

TL;DR

This paper studies the stability properties of parabolic equations in non-cylindrical domains, revealing that degeneracy can improve solution estimates and that solutions may not decay exponentially as in cylindrical domains.

Contribution

It provides new insights into the stability of parabolic equations in non-cylindrical domains, highlighting the positive effects of degeneracy on solution estimates.

Findings

Degeneracy positively impacts $L^ Infty$-norm estimates.

Solutions may not decay exponentially in non-cylindrical domains.

The stability analysis covers both nondegenerate and degenerate cases.

Abstract

This paper addresses the stability of a class of parabolic equations in non-cylindrical domains. We investigate the $L^\infty$-stability of systems for both nondegenerate and degenerate cases. Unlike in cylindrical domains, solutions to such problems may not exhibit exponential decay. An interesting phenomenon observed is that degeneracy has a positive impact on $L^\infty$-norm estimates for solutions to the system.