Strong Global Attractors for 3D Wave Equations with Weakly Damping

Abstract

We consider the existence of the global attractor ${\mathcal{A}}_1$ for the 3D weakly damped wave equation. We prove that ${\mathcal{A}}_1$ is compact in $\left(H^2\right(\Omega )\cap H_0^1(\Omega \left)\right)\times H_0^1\left(\Omega \right)$ and attracts all bounded subsets of $\left(H^2\right(\Omega )\cap H_0^1(\Omega \left)\right)\times H_0^1\left(\Omega \right)$ with respect to the norm of $\left(H^2\right(\Omega )\cap H_0^1(\Omega \left)\right)\times H_0^1\left(\Omega \right)$. Furthermore, this attractor coincides with the global attractor in the weak energy space $H_0^1\left(\Omega \right)\times L^2\left(\Omega \right)$.