Damped wave equation with super critical nonlinearities

Abstract

We study global existence in time of small solutions to the Cauchy problem for the nonlinear damped wave equation \begin{equation} \left\{ \begin{array}{c} \partial _{t}^{2}u+\partial _{t}u-\Delta u=\mathcal{N}\left( u\right) ,\quad 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) , \quad x\in \mathbf{R}^{n}, \end{array} \right. \tag*{(0.1)} \end{equation} where $\varepsilon >0$. The nonlinearity $\mathcal{N}\left( u\right) \in \mathbf{C}^{k}\left( \mathbf{R}\right) $ satisfies the estimate \begin{equation*} \Big | \frac{d^{j}}{du^{j}}\mathcal{N}\left( u\right) \Big | \leq C\big | u\big | ^{\rho -j}, \quad 0\leq j\leq k\leq \rho . \end{equation*} The power $\rho >1+\frac{2}{n}$ is considered as super critical for large time. We assume that the initial data \begin{equation*} u_{0}\in \mathbf{H}^{\alpha ,0}\cap \mathbf{H}^{0,\delta },\text{ }u_{1}\in \mathbf{H}^{\alpha -1,0}\cap \mathbf{H}^{0,\delta }, \end{equation*} where $\delta >\frac{n}{2},$ $\left[ \alpha \right] \leq \rho ,\alpha \geq \frac{n}{2}-\frac{1}{\rho -1}$ for $n\geq 2$, and $\alpha \geq \frac{1}{2}- \frac{1}{2\left( \rho -1\right) }$ for $n=1.$ Weighted Sobolev spaces are \begin{equation*} \mathbf{H}^{l,m}= \Big \{ \phi \in \mathbf{L}^{2}; \big \| \left\langle x\right\rangle ^{m}\left\langle i\partial _{x}\right\rangle ^{l}\phi \left( x\right) \big \| _{\mathbf{L}^{2}}s < \infty \Big \} , \end{equation*} where $\left\langle x\right\rangle =\sqrt{1+x^{2}}.$ Then we prove that there exists a small $\varepsilon _{0}>0$ such that for any $\varepsilon \in \left( 0,\varepsilon _{0}\right] $ there exists a unique global solution $ u\in \mathbf{C}\left( \left[ 0,\infty \right) ;\mathbf{H}^{\alpha ,0}\cap \mathbf{H}^{0,\delta }\right) $ for the Cauchy problem (0.1) and solutions satisfy the time decay property \begin{equation*} \big \| u ( t ) \big \| _{\mathbf{L}^{p}}\leq Ct^{-\frac{n}{2 } ( 1-\frac{1}{p} ) } \end{equation*} for all $t>0$, where $2\leq p\leq $ $\frac{2n}{n-2\alpha }$ if $\alpha s < \frac{n}{2},$ $2\leq ps <$ $\infty $ if $\alpha =\frac{n}{2},$ and $2\leq p\leq $ $\infty $ if $\alpha \gt\frac{n}{2}.$