JournalszaaVol. 31, No. 3pp. 267–282

Diffusion Phenomenon for Linear Dissipative Wave Equations

  • Belkacem Said-Houari

    King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia
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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 2p2 \leq p\leq \infty ,\, uvLp(RN)\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 vv, where uu is the solution of the linear wave equation with initial data (u0,u1)(H1(RN)L1,γ(RN))×(L2(RN)L1,γ(RN))\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 γ[0,1]\gamma \in \left[ 0,1\right] , and vv 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 tγ2t^{-\frac{\gamma}{2}}.

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