Solution Representations for a Wave Equation with Weak Dissipation

2002/10/02 by Jens Wirth · 3 citations
Mathematics · #math.AP #msc:35L05 #msc:35L15 #msc:35B45

paper · pdf

published as Math. Meth. Appl. Sci. 27/1 (2004) 101-124 · 29 pages, 2 figures

arxiv created 2002/10/02 · arxiv updated 2009/11/30

Abstract

We consider the Cauchy problem for the weakly dissipative wave equation \bx v+\fracμ1+tv_t=0, x∈\Rⁿ, t≥ 0, parameterized by μ>0, and prove a representation theorem for its solution using the theory of special functions. This representation is used to obtain L_p--L_q estimates for the solution and for the energy operator corresponding to this Cauchy problem. Especially for the L₂ energy estimate we determine the part of the phase sp which is responsible for the decay rate. It will be shown that the situation d strongly on the value of μ and that μ=2 is critical.

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