Wave equations with time-dependent dissipation I. Non-effective dissipation

Abstract

The goal of this article is to construct structural representations of the solutions to Cauchy problems for weakly dissipative wave equations below scaling and to deduce estimates of the solution and its energy based on L q (R n ), q 2. Furthermore, the sharpness of the obtained estimates is discussed. IntroductionCauchy problems for wave equations with weak dissipationis assumed to be positive and tends to zero as t tends to infinity, provide an important model problem for the study of asymptotic behaviours and the influence of lower-order terms on them. As usual we denote D = −i* and = j * 2 x j . We refer to [RW05] for an exposition of results and for the classification of * Fax: +49373 139442.

The threshold of effective damping for semilinear wave equations

“…The special case µ = 2 + n(2/m − 1) can be easily proved by replacing estimate (40) with (41), whereas estimates (26) and (27) follow from (36) and (37).…”

Wave equations with time-dependent dissipation II. Effective dissipation

“…We refer to [13] for an exposition of results and for the classification of time-dependent dissipation terms. In [18] it was shown that, roughly speaking, under the non-effectivity assumption b(t) = O(t −1 ) E-mail address: wirth@math.tu-freiberg.de. solutions are closely related to free waves.…”

Fourier Analysis

“…We refer the reader to [8,9]. On the other hand, if β > 1 then the asymptotic profile of the solution of (1.2) is given by that of the free wave equation w = 0 (see [13]). Wirth [12] considered the linear problem…”