Hyperbolic relaxation of reaction-diffusion equations with dynamic boundary conditions

Under consideration is the hyperbolic relaxation of a semilinear reaction-diffusion equation \begin{equation*} \varepsilon u_{tt}+u_{t}-\Delta u+f(u)=0 \end{equation*} on a bounded domain $\Omega \subset \mathbb {R}^{3}$ with $\varepsilon \in (0,1]$ and the prescribed dynamic condition \begin{equation*} \partial _{\mathbf {n}}u+u+u_{t}=0 \end{equation*} on the boundary $\Gamma :=\partial \Omega$. We also consider the limit parabolic problem ($\varepsilon =0$) with the same dynamic boundary condition. Each problem is well-posed in a suitable phase space where the global weak solutions generate a Lipschitz continuous semiflow which admits a bounded absorbing set. Because of the nature of the boundary condition, fractional powers of the Laplace operator are not well-defined. The precompactness property required by the hyperbolic semiflows for the existence of the global attractors is gained through the approach of Pata and Zelik (2006). In this case, the optimal regularity for the global attractors is also readily established. In the parabolic setting, the regularity of the global attractor is necessary for the semicontinuity result. After fitting both problems into a common framework, a proof of the upper-semicontinuity of the family of global attractors is given at $\varepsilon =0$. Finally, we also establish the existence of a family of exponential attractors.