Damped wave equation with a critical nonlinearity

Abstract:

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity \begin{equation*} \left \{ \begin {array}{c} \partial _{t}^{2}u+\partial _{t}u-\Delta u+\lambda u^{1+\frac {2}{n}}=0,\text { }x\in \mathbf {R}^{n},\text { }t>0, u(0,x)=\varepsilon u_{0}\left ( x\right ) ,\partial _{t}u(0,x)=\varepsilon u_{1}\left ( x\right ) ,x\in \mathbf {R}^{n}, \end{array} \right . \end{equation*} where $\varepsilon >0,$ and space dimensions $n=1,2,3$. Assume that the initial data \begin{equation*} u_{0}\in \mathbf {H}^{\delta ,0}\cap \mathbf {H}^{0,\delta },\text { }u_{1}\in \mathbf {H}^{\delta -1,0}\cap \mathbf {H}^{-1,\delta }, \end{equation*} where $\delta >\frac {n}{2},$ weighted Sobolev spaces are \begin{equation*} \mathbf {H}^{l,m}=\left \{ \phi \in \mathbf {L}^{2};\left \Vert \left \langle x\right \rangle ^{m}\left \langle i\partial _{x}\right \rangle ^{l}\phi \left ( x\right ) \right \Vert _{\mathbf {L}^{2}}<\infty \right \} , \end{equation*} $\left \langle x\right \rangle =\sqrt {1+x^{2}}.$ Also we suppose that \begin{equation*} \lambda \theta ^{\frac {2}{n}}>0,\int u_{0}\left ( x\right ) dx>0, \end{equation*} where \begin{equation*} \text { }\theta =\int \left ( u_{0}\left ( x\right ) +u_{1}\left ( x\right ) \right ) dx\text {.} \end{equation*} 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{equation*} \left \Vert u\left ( t\right ) -\varepsilon \theta G\left ( t,x\right ) e^{-\varphi \left ( t\right ) }\right \Vert _{\mathbf {L}^{p}}\leq C\varepsilon ^{1+\frac {2}{n}}g^{-1-\frac {n}{2}}\left ( t\right ) \left \langle t\right \rangle ^{-\frac {n}{2}\left ( 1-\frac {1}{p}\right ) } \end{equation*} for all $t>0,$ $1\leq p\leq \infty ,$ where $\varepsilon \in \left ( 0,\varepsilon _{0}\right ] .$