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 $3\le d\le 5$. 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.

References in corpus (1)

• $L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

Cited by in corpus (2)

• Non relativistic and ultra relativistic limits in 2d stochastic nonlinear damped Klein-Gordon equation
• Damped Strichartz estimates and the incompressible Euler--Maxwell system