Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data

Abstract

This paper is concerned with weighted energy estimates for solutions to wave equation ∂ 2 t u − ∆u + a(x)∂ t u = 0 with space-dependent damping term a(x) = |x| −α (α ∈ [0, 1)) in an exterior domain Ω having a smooth boundary. The main result asserts that the weighted energy estimates with weight function like polymonials are given and these decay rate are almost sharp, even when the initial data do not have compact support in Ω. The crucial idea is to use special solution of ∂ t u = |x| α ∆u including Kummer's confluent hypergeometric functions. (2010): Primary: 35L20 , Secondary: 35B40, 47B25. Mathematics Subject Classification

On global existence for semilinear wave equations with spacedependent critical damping

“…The idea of such a decomposition is based on Sobajima [26] and Ikehata-Sobajima [13] (motivated by so-called modified Morawetz method in Ikehata-Matsuyama [12]). The weighted estimates for v (and ∂ t v) are valid via a weighted energy estimate due to Sobajima-Wakasugi [27]. Then combining these estimates and energy estimates for 0 < V 0 < N − 1, we can deduce the desired energy estimates for the case V 0 > N − 1.…”

Critical exponent for the semilinear wave equations with a damping increasing in the far field

“…The space-dependent damping α ∈ R, β = 0 was also studied by [53,42,21,60,25,65,55,56,57,58], and similarly to the above, the behavior of the solution was classified in the following way: (i) Scattering: if α > 1, then the solution behaves like that of the wave equation without damping; (ii) Scale-invariant weak damping: if α = 1, then the asymptotic behavior of the solution depends on a 0 ; (iii) Effective: if α < 1, then the solution behaves like that of the corresponding parabolic equation. We note that in the space-dependent case, the overdamping phenomenon does not occur.…”

Global existence of solutions to semilinear damped wave equation with slowly decaying inital data in exterior domain

“…for fixed λ ≥ 0, by using the idea of weighted energy estimates including Kummer's confluent hypergeometric functions, originated in [25].…”

“…Recently, Wakasugi-Sobajima [25] found a framework of weighted energy estimates with a weight function of polynomial type. In [25], the weight function is taken as the inverse of the positive solution of heat equation ∂ t Φ = ∆Φ including the Kummer confluent hypergeometric function (see Section 2.1 below). This enables us to obtain the weighted energy estimate of polynomial type.…”

“…The purpose of the present paper is to discuss the nonlinear problem of damped wave equation in exterior domain in view of weighted energy estimate of polynomial type introduced by [25].…”