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Damped wave equation with a critical nonlinearity

Authors:	Nakao Hayashi Elena I Kaikina Pavel I Naumkin

Institution:	Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka, 560-0043, Japan ; Departamento de Ciencias Básicas, Instituto Tecnológico de Morelia, Morelia CP 58120, Michoacán, Mexico ; Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico

Abstract:	We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity $\begin{displaymath}\left\{ \begin{array}{c} \partial _{t}^{2}u+\partial _{t}u-\... ...u_{1}\left( x\right) ,x\in \mathbf{R}^{n}, \end{array}\right. \end{displaymath}$ where $\varepsilon >0,$ and space dimensions . Assume that the initial data $\begin{displaymath}u_{0}\in \mathbf{H}^{\delta ,0}\cap \mathbf{H}^{0,\delta },\t... ..._{1}\in \mathbf{H}^{\delta -1,0}\cap \mathbf{H}^{-1,\delta }, \end{displaymath}$ where $\delta >\frac{n}{2},$ weighted Sobolev spaces are $\begin{displaymath}\mathbf{H}^{l,m}=\left\{ \phi \in \mathbf{L}^{2};\left\Vert \... ...left( x\right) \right\Vert _{\mathbf{L}^{2}}<\infty \right\} , \end{displaymath}$ $\left\langle x\right\rangle =\sqrt{1+x^{2}}.$ Also we suppose that $\begin{displaymath}\lambda \theta ^{\frac{2}{n}}>0,\int u_{0}\left( x\right) dx>0, \end{displaymath}$ where $\begin{displaymath}\text{ }\theta =\int \left( u_{0}\left( x\right) +u_{1}\left( x\right) \right) dx\text{.} \end{displaymath}$ Then we prove that there exists a positive $\varepsilon _{0}$ such that the Cauchy problem above has a unique global solution $u\in \mathbf{C}\left( \left 0,\infty \right) ;\mathbf{H}^{\delta ,0}\right)$ satisfying the time decay property $\begin{displaymath}\left\Vert u\left( t\right) -\varepsilon \theta G\left( t,x\r... ...gle t\right\rangle ^{-\frac{n}{2}\left( 1-\frac{1}{p}\right) } \end{displaymath}$ for all $1\leq p\leq \infty ,$ where $\varepsilon \in \left( 0,\varepsilon _{0}\right] .$

Keywords:	Damped wave equation large time asymptotics

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