Global asymptotics for the damped wave equation with absorption in higher dimensional space

Abstract

We consider the Cauchy problem for the damped wave equation with absorption$$u_{tt}-\mathrm{\Delta }u+u_t+|u{|}^{\rho -1}u=0,(t,x)\in {\mathbf{R}}_{+}\times {\mathbf{R}}^N,(*)$$with $N=3,4$. The behavior of $u$ as $t\to \mathrm{\infty }$ is expected to be the Gauss kernel in the supercritical case $\rho >{\rho }_c\left(N\right):=1+2/N$. In fact, this has been shown by Karch [12] (Studia Math., 143 (2000), 175--197) for $\rho >1+\frac{4}{N}(N=1,2,3)$, Hayashi, Kaikina and Naumkin [8] (preprint (2004)) for $\rho >{\rho }_c\left(N\right)(N=1)$ and by Ikehata, Nishihara and Zhao [11] (J. Math. Anal. Appl., 313 (2006), 598--610) for and . Developing their result, we will show the behavior of solutions for , . For the proof, both the weighted $L^2$-energy method with an improved weight developed in Todorova and Yordanov [22] (J. Differential Equations, 174 (2001), 464--489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for $N=5$, because the semilinear term is not in $C^2$ and the second derivatives are necessary when the explicit formula of solutions is estimated.