Dynamics for the heat equation with memory and Hardy potentials

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 $ \left \{ S_{\lambda }\left ( t\right )\right \}_{t \ge 0} $ for every $ \lambda \in \left [0, \lambda ^{*} \right ) $ on $ L^2(\Omega)\times L_\mu^2(\mathbb{R}_+; H_0^1(\Omega)) $ were defined. Second, the existence of global attractors $ \mathcal{A}_{\lambda } $ for every $ \lambda \in \left [0, \lambda ^{*} \right ) $ was proved by verifying the asymptotic compactness of the semigroups $ \left \{ S_{\lambda }\left ( t\right )\right \}_{t \ge 0} $ for any $ \lambda \in \left [0, \lambda ^{*} \right ) $ (Theorem 4.3). Finally, we further showed that the attractors $ \mathcal{A}_{\lambda } $ for $ \lambda \in \left [0, \lambda ^{*} \right ) $ are uniformly bounded (Theorem 5.1) and upper semicontinuous at $ \lambda = 0 $ (Theorem 5.4).