Stability of parabolic equations in non-cylindrical domains

This paper addresses the stability of a class of parabolic equations in non-cylindrical domains. We investigate the $L^{\mathrm{\infty }}$-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 $L^{\mathrm{\infty }}$-norm estimates for solutions to the system.