Dynamics for the heat equation with memory and Hardy potentials

Fei Wang, Shan Ma, Chunyu Qiu

Abstract: This paper aims to investigate the heat equation with both fading memory and Hardy potential $\frac{\lambda}{|x|^2}$ in bounded domains containing the origin. First, we established the well-posedness of weak solutions (Theorem 3.2), and then the solution semigroups ${S_\lambda (t)\}_{t \ge 0}$ for every $\lambda \in [0, \lambda^*)$ on $L^2(\Omega) \times L^2_\mu(\mathbb{R}_+; H^1_0(\Omega))$ were defined. Second, the existence of global attractors $\mathcal{A}_\lambda$ for every $\lambda \in [0, \lambda^*)$ was proved by verifying the asymptotic compactness of the semigroups $\{S_\lambda (t)\}_{t \ge 0}$ for any $\lambda \in [0, \lambda^*)$ (Theorem 4.3). Finally, we further showed that the attractors $\mathcal{A}_\lambda$ for $\lambda \in [0, \lambda^*)$ are uniformly bounded (Theorem 5.1) and upper semicontinuous at $\lambda = 0$ (Theorem 5.4).