$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data
Masahiro Ikeda Takahisa Inui Mamoru Okamoto Yuta Wakasugi
Communications on Pure & Applied Analysis 2019, 18(4): 1967-2008 doi: 10.3934/cpaa.2019090
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) $
$ 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.
keywords: Nonlinear damped wave equation $ L^p $-$ L^q $ estimates critical exponent

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