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Damped wave equation with a critical nonlinearity

Author(s): Nakao Hayashi; Elena I. Kaikina; Pavel I. Naumkin
Journal: Trans. Amer. Math. Soc. 358 (2006), 1165-1185.
MSC (2000): Primary 35Q55; Secondary 35B40
Posted: April 22, 2005
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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 $n=1,2,3$. 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 $t>0,$ $1\leq p\leq \infty ,$ where $\varepsilon \in \left( 0,\varepsilon _{0}\right] .$

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Additional Information:

Nakao Hayashi
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka, 560-0043, Japan
Email: nhayashi@math.wani.osaka-u.ac.jp

Elena I. Kaikina
Affiliation: Departamento de Ciencias Básicas, Instituto Tecnológico de Morelia, Morelia CP 58120, Michoacán, Mexico
Email: ekaikina@matmor.unam.mx

Pavel I. Naumkin
Affiliation: Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico
Email: pavelni@matmor.unam.mx

DOI: 10.1090/S0002-9947-05-03818-3
PII: S 0002-9947(05)03818-3
Keywords: Damped wave equation, large time asymptotics
Received by editor(s): April 1, 2003
Received by editor(s) in revised form: April 22, 2004
Posted: April 22, 2005
Additional Notes: The second and the third authors were supported in part by CONACYT
Copyright of article: Copyright 2005, American Mathematical Society

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