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

Abstract

We study the Cauchy problem of the damped wave equation \begin{align*} \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1\le q \le p < \infty\ (p\neq 1)$ with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in $(H^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r)$ with $r \in (1,2]$, $s\ge 0$, and $\beta = (n-1)|\frac{1}{2}-\frac{1}{r}|$, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power $1+\frac{2r}{n}$, while it is known that the critical power $1+\frac{2}{n}$ belongs to the blow-up region when $r=1$. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan and blow-up rates by an ODE argument.