I am interested in the following IBVP for the strongly damped wave equation: \begin{equation} u_{tt}-c^2\Delta u-b\Delta u_t+eu_t=f(x,t) \quad \text{in} \ \Omega \times (0,T), \\ u=0 \quad \text{on} \ \partial \Omega, \\ (u,u_t)_{\vert t=0}=(u_0,u_1), \end{equation} where $c^2, b, e>0$. I would like to have (under appropriate assumptions on the initial data and $f$) the weak solution which is $L^\infty$-regular in space. In other words, I would like to work in the space $H_0^1(\Omega) \cap L^\infty(\Omega)$ instead of the usual $H_0^1(\Omega)$.
Is a result of this type available in the literature? Any help would be appreciated.
