American Institute of Mathematical Sciences

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 by an ODE argument.