Stability of parabolic equations in non-cylindrical domains

TL;DR

This work analyzes $L^\infty$-stability of parabolic equations in non-cylindrical domains with moving boundaries and a possibly degenerate diffusion coefficient $a(x)=x^\alpha$. Using weighted multiplier methods, Hardy inequalities, spectral analysis, and comparison principles, it derives precise decay rates depending on the degeneracy parameter $\alpha$ and the boundary-growth exponent $\gamma$ for both nondegenerate ($\alpha=0$) and degenerate ($0<\alpha<1$) cases. The results reveal a sharp stability taxonomy: exponential, subexponential, or polynomial decay, with degeneracy improving stability at the critical threshold (e.g., $\gamma=1/2$ in the nondegenerate vs degenerate settings) and explicit rates at the transition points. These findings illuminate how moving boundaries interact with diffusion strength to dictate long-time behavior, with potential implications for diffusion processes in domains with time-dependent geometry and degenerate media.

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.

Theorems & Definitions (18)

• Definition 1.1
• Remark 1.1
• Theorem 1.1
• Remark 1.2
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 2.1
• Lemma 2.1
• Proposition 2.1