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Zeitschrift für Analysis und ihre Anwendungen


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Volume 31, Issue 3, 2012, pp. 267–282
DOI: 10.4171/ZAA/1459

Published online: 2012-07-11

Diffusion Phenomenon for Linear Dissipative Wave Equations

Belkacem Said-Houari[1]

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

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 \leq p\leq \infty $,\, $\left\Vert u-v\right\Vert _{L^{p }(\mathbb{R}^{N})}$ decays with the rate $ %%t^{-\left(\!\frac{N}{2}(1-\frac{1}{p})\!\right)-1-\frac{\gamma}{2}}, t^{-\frac{N}{2}(1-\frac{1}{p})-1-\frac{\gamma}{2}}, \,\gamma \in \lbrack 0,1]$ faster than that of either $% u $ or $v$, where $u$ is the solution of the linear wave equation with initial data $\left( u_{0},u_{1}\right) \in \left( H^{1}(\mathbb{R}^{N})\cap L^{1,\gamma }(\mathbb{R}^{N})\right) \times \left( L^{2}(\mathbb{R}^{N})\cap L^{1,\gamma }(\mathbb{R}^{N})\right) $ with $\gamma \in \left[ 0,1\right] $, 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 $t^{-\frac{\gamma}{2}}$.

Keywords: Diffusion phenomenon, Cauchy problem, damped wave equation, heat equation, asymptotic behavior

Said-Houari Belkacem: Diffusion Phenomenon for Linear Dissipative Wave Equations. Z. Anal. Anwend. 31 (2012), 267-282. doi: 10.4171/ZAA/1459