# Diffusion Phenomenon for Linear Dissipative Wave Equations

### Belkacem Said-Houari

King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia

## Abstract

In this paper we prove the diffusion phenomenon for the linear wave equation. To derive the diffusion phenomenon, a new method is used. In fact, for initial data in some weighted spaces, prove that for $2≤p≤∞$ , $∥u−v∥_{L_{p}(R_{N})}$ decays with the rate $t_{−2N(1−p1)−1−2γ}$, $γ∈[0,1]$ faster than that of either $u$ or $v$, where $u$ is the solution of the linear wave equation with initial data $(u_{0},u_{1})∈(H_{1}(R_{N})∩L_{1,γ}(R_{N}))×(L_{2}(R_{N})∩L_{1,γ}(R_{N}))$ with $γ∈[0,1]$ , and $v$ is the solution of the related heat equation with initial data $v_{0}=u_{0}+u_{1}$. This result improves the result in H. Yang and A. Milani [Bull. Sci. Math. 124 (2000), 415–433] in the sense that, under the above restriction on the initial data, the decay rate given in that paper can be improved by $t_{−2γ}$.

## Cite this article

Belkacem Said-Houari, Diffusion Phenomenon for Linear Dissipative Wave Equations. Z. Anal. Anwend. 31 (2012), no. 3, pp. 267–282

DOI 10.4171/ZAA/1459