Lp-Lq estimates for damped wave equations and their applications to semi-linear problem

Abstract

In this paper we study the Cauchy problem to the linear damped wave equation $u_{tt}-\Delta u+2a{u}_t=0$ in $(0,\infty )$$\times R^n(n\ge 2)$. It has been asserted that the above equation has the diffusive structure as $t\to \infty$. We give the precise interpolation of the diffusive structure, which is shown by $L^p\text{-}{L}^q$ estimates. We apply the above $L^p\text{-}{L}^q$ estimates to the Cauchy problem for the semilinear damped wave equation $u_{tt}-\Delta u+$$2a{u}_t=|u{|}^{\sigma }u$ in $(0,\infty )$$\times R^n(2\le n\le 5)$. If the power $\sigma$ is larger than the critical exponent $2/n$(Fujita critical exponent) and it satisfies $\sigma \le 2/(n-2)$ when $n\ge 3$, then the time global existence of small solution is proved, and the decay estimates of several norms of the solution are derived.