Asymptotic dynamics of non-autonomous fractional reaction-diffusion equations on bounded domains

Abstract

In this paper, we consider the asymptotic dynamics of non-autonomous fractional reaction-diffusion equations of the form \[u_{t}+(-\Delta)^{s}u+f(u)=g(t)\] complemented with the Dirichlet boundary condition on a bounded domain. First, we obtain some higher-attraction results for pullback attractors, that is, without any additional $t$-differentiability assumption on the forcing term $g$, for any space dimension $N$ and any growth power $p\geq2$ of $f$, the known $(L^{2}(\Omega), L^{2}(\Omega))$ pullback attractor can indeed attract every $L^{2}(\Omega)$-bounded set in the $L^{2+\delta}(\Omega)$-norm for every $\delta\in [0,\infty)$ as well as in the $W_{0}^{s,2}(\Omega)$-norm. Then, we construct a family of Borel probability measures $\{\mu_{t}\}_{t\in\mathbb{R}}$, whose supports satisfy the higher-attraction results. Finally, we investigate the relationship between such the Borel probability measures and time-dependent statistical solutions for this fractional Laplacian equation.