Diffusion Phenomenon for Linear Dissipative Wave Equations

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 $2 \leq p\leq \infty$ ,\, $\left\Vert u-v\right\Vert _{L^{p }(\mathbb{R}^{N})}$ decays with the rate $$ faster than that of either $$ 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 $$. 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 $t^{-\frac{\gamma}{2}}$.