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

Michael Reissig·Jens Wirth
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Abstract

This expository article is intended to give an overview about recently achieved results on asymptotic properties of solutions to the Cauchy problem $u_{tt}-Δu+b(t)u_t =0,\qquad u(0,\cdot)=u_1,\quad \mathrm{D}_tu(0,\cdot)=u_2$ 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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Michael Reissig, Jens Wirth. 2005-08-28. $L^p$--$L^q$ decay estimates for wave equations with monotone time-dependent dissipation. https://arxiv.org/abs/math/0508549

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