Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain

Abstract

Abstract. This paper is concerned with weighted energy estimates and diffusion phenomena for the initial-boundary problem of the wave equation with space-dependent damping term in an exterior domain. In this analysis, an elliptic problem was introduced by Todorova and Yordanov. This attempt was quite useful when the coefficient of the damping term is radially symmetric. In this paper, by modifying their elliptic problem, we establish weighted energy estimates and diffusion phenomena even when the coefficient of the damping term is not radially symmetric.

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

“…As in [28] (see also [23] and [24]), we have By (5.4), we have J 1 ≤ K 1 (1 + t) − γ−α 2(2−α) (E 0 + E 1 ) 1 2 . By a computation similar to the one for J 1 , we deduce from (5.3) that…”

“…Then in [23,24] the problem (1.1) in an exterior domain with non-radially symmetric damping terms satisfying lim |x|→∞ |x| α a(x) = a 0 > 0 could be considered and it is shown that the asymptotic behavior of solutions to (1.1) can be also given by the solution of (2.1), however, only when the initial data are compactly supported. We would summarize that if the initial data are not compactly supported, then a kind of weighted energy estimates is quite few; note that Ikehata gave one of weighted energy estimates in [8] but the initial data are required to have an exponential decay.…”

“…This paper is organized as follows. In Section 2, we construct a suitable weight function from a selfsimilar solution of (2.1) with a parameter, which is quite different from that in [8], [26], [21], [28], [23] and [24] as mentioned before. We also mention the properties of the semigroup generated by L * = a(x)∆.…”

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

“…Theorem 1.2 is proved in Section 2 by weighted energy method with a weight function having the form e ψ(t,x) with an appropriate function ψ(t, x) (see Definition 2.1). Such a weight function was developed by [21,60,48,56]. Making use of this weight, we can estimate the weighted energy of the solution by the sum of the initial energy and the nonlinear terms (see Lemma 2.4).…”

“…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.…”