The Strichartz estimates for the damped wave equation and the behavior of solutions for the energy critical nonlinear equation

Abstract

For the linear damped wave equation (DW), the $L^p$-$L^q$ type estimates have been well studied. Recently, Watanabe showed the Strichartz estimates for DW when $d=2,3$. In the present paper, we give Strichartz estimates for DW in higher dimensions. Moreover, by applying the estimates, we give the local well-posedness of the energy critical nonlinear damped wave equation (NLDW) $\partial_t^2 u - \Delta u +\partial_t u = |u|^{\frac{4}{d-2}}u$, $(t,x) \in [0,T) \times \mathbb{R}^d$, where $3 \leq d \leq 5$. Especially, we show the small data global existence for NLDW. In addition, we investigate the behavior of the solutions to NLDW. Namely, we give a decay result for solutions with finite Strichartz norm and a blow-up result for solutions with negative Nehari functional.