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 $2\le p\le \infty$ , ${\Vert u-v\Vert }_{L^p({\mathbb{R}}^N)}$ decays with the rate $t^{-\frac{N}{2}(1-\frac{1}{p})-1-\frac{\gamma }{2}}$, $\gamma \in [0,1]$ 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 $t^{-\frac{\gamma }{2}}$.