$L^p$--$L^q$ decay estimates for wave equations with monotone time-dependent dissipation

Details $L^p$--$L^q$ decay estimates for wave equations with monotone time-dependent dissipation $L^p$--$L^q$ decay estimates for wave equations with monotone time-dependent dissipation

2005-08-28

arxiv.org/abs/math/0508549v1

RIMS Kokyuroku Nr. 1475, p. 91-106, Kyoto University, 2006

Mathematics

Analysis of PDEs

15 pages, 2 figures

Scientific paper

This expository article is intended to give an overview about recently achieved results on asymptotic properties of solutions to the Cauchy problem $u_{tt}-\Delta u+b(t)u_t =0,\qquad u(0,\cdot)=u_1,\quad \mathrm{D}_tu(0,\cdot)=u_2$ for a wave equation with time-dependent dissipation term. The results are based on structural properties of the Fourier multipliers representing its solution. The article explains the general philosophy behind the approach.

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