| Contents |
| We consider Dirichlet problem for a semi-linear wave equation with damping term $(-\Delta)^\alpha\partial_t u$, where $\alpha\in[0,\frac{1}{2}]$, in a bounded smooth domain $\Omega\subset \Bbb R^3$, assuming initial data from usual energy space $H^1_0(\Omega)\times L^2(\Omega)$. First, we establish control of $L^5([0,T];L^{10}(\Omega))$ norm of solutions for corresponding linear non-autonomous problem in terms of energy norm, which does not follow from energy estimate as well as Strichartz estimates for pure wave equation. Then treating semi-linear equation as perturbation of the linear problem we establish its well-posedness in the class of energy solutions with finite $L_{loc}^5(\Bbb R_{+}, L^{10}(\Omega ))$ norm. Moreover, we show that solutions from the mentioned class possess smoothing property analogous to solutions of parabolic equation. Finally we show that dynamical system generated by these solutions possesses a smooth global attractor. |
|