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 ,\, decays with the rate faster than that of either or , where is the solution of the linear wave equation with initial data with , and 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 .
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Belkacem Said-Houari, Diffusion Phenomenon for Linear Dissipative Wave Equations. Z. Anal. Anwend. 31 (2012), no. 3, pp. 267–282DOI 10.4171/ZAA/1459