$ 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

doi:10.3934/cpaa.2019090

Article Contents

2019, Volume 18, Issue 4: 1967-2008. Doi: 10.3934/cpaa.2019090

This issue Previous Article Spike layer solutions for a singularly perturbed Neumann problem: Variational construction without a nondegeneracy Next Article Existence, multiplicity and concentration for a class of fractional

$ p \& q $

Laplacian problems in

$ \mathbb{R} ^{N} $

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

1.
Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan
2.
Center for Advanced Intelligence Project, RIKEN, Japan
3.
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
4.
Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553, Japan
5.
Department of Engineering for Production and Environment, Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime, 790-8577, Japan

Received: August 2018

Revised: October 2018

Published: January 2019

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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 by an ODE argument.

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Mathematics Subject Classification: Primary: 35L71; Secondary: 35A01, 35B40, 35B44.

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