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 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