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ISSN 1088-6850(e) ISSN 0002-9947(p)

Damped wave equation with a critical nonlinearity

Authors: Nakao Hayashi, Elena I. Kaikina and Pavel I. Naumkin
Journal: Trans. Amer. Math. Soc. 358 (2006), 1165-1185
MSC (2000): Primary 35Q55; Secondary 35B40
Posted: April 22, 2005
MathSciNet review: 2187649
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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 $\varepsilon \in \left( 0,\varepsilon _{0}\right] .$

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