E. A. Tavares--Lima; B. Lorenzi

Universidade de São Paulo

Title: Asymptotic behavior of parabolic equations in degenerating thin domains

Abstract: In this work, we study the asymptotic behavior of a family of semilinear parabolic problems defined on thin domains $R_\varepsilon \subset \mathbb{R} ^{1+n}$, whose geometry is determined by a nonnegative function vanishing at an endpoint.

By performing a suitable change of variables, we reformulate the problem in a fixed reference domain and derive uniform estimates for the solutions. We first establish the convergence of equilibria and obtain explicit rates of convergence. Building upon these results, we analyze the associated parabolic dynamics and prove the convergence of the global attractors with an explicit rate as $\varepsilon \to 0$.