Stability of strong global and exponential attractors for semilinear beam equations with fractional damping and memory

Abstract

This paper investigates the stability of strong global and exponential attractors for a semilinear beam equation with memory and fractional damping, where $\alpha \in [0,2]$ denotes the fractional damping exponent and $\beta \in [0,1]$ the memory parameter. After showing the existence of a strong global attractor, we prove its upper semicontinuity in the parameter pair $(\alpha ,\beta )$. We then construct a family of strong exponential attractors and establish its continuity in $(\alpha ,\beta )$. Here, ``strong" means that the compactness, attraction, and parameter-robustness properties are established in a topology stronger than that of the phase space. Compared to existing $\beta =0$ results, our findings hold in a stronger topology. The analysis draws upon our recent higher-order regularity results for global attractors.