Solution representations for a wave equation with weak dissipation

Abstract

SUMMARYWe consider the Cauchy problem for the weakly dissipative wave equationparameterized by ¿0, and prove a representation theorem for its solutions using the theory of special functions. This representation is used to obtain Lp-Lq estimates for the solution and for the energy operator corresponding to this Cauchy problem.Especially for the L 2 energy estimate we determine the part of the phase space which is responsible for the decay rate. It will be shown that the situation depends strongly on the value of and that = 2 is critical.

The threshold of effective damping for semilinear wave equations

“…To prove them we follow the approach used in [25] to derive L 2 − L 2 estimates for the linear damped wave equation…”

A shift in the Strauss exponent for semilinear wave equations with a not effective damping

“…The damping term of this model is not effective (see [27]). Nevertheless, there should be an improving influence on the critical exponent p crit in comparison with Strauss exponent p 0 (n) for      w tt − △w = |w| p , t ≥ 0, x ∈ R n , w(0, x) = w 0 (x), x ∈ R n , w t (0, x) = w 1 (x), x ∈ R n ,…”

“…It has been recently shown that the critical exponent for models with effective dissipation, this means, µ is sufficiently large (see [27])…”

Fourier Analysis

“…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…”