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

We study the Cauchy problem of the damped wave equation \begin{document}$ \begin{align*} \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*} $\end{document} and give sharp \begin{document}$ L^p $\end{document} - \begin{document}$ L^q $\end{document} estimates of the solution for \begin{document}$ 1\le q \le p 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 \begin{document}$ (H^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r) $\end{document} with \begin{document}$ r \in (1,2] $\end{document} , \begin{document}$ s\ge 0 $\end{document} , and \begin{document}$ \beta = (n-1)|\frac{1}{2}-\frac{1}{r}| $\end{document} , and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power \begin{document}$ 1+\frac{2r}{n} $\end{document} , while it is known that the critical power \begin{document}$ 1+\frac{2}{n} $\end{document} belongs to the blow-up region when \begin{document}$ r = 1 $\end{document} . 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 by an ODE argument.