Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain

Abstract

Abstract. In this paper, we consider the asymptotic behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We prove that when the damping is effective, the solution is approximated by that of the corresponding heat equation as time tends to infinity. Our proof is based on semigroup estimates for the corresponding heat equation and weighted energy estimates for the damped wave equation. The optimality of the decay late for solutions is also established.

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

“…−1 ∆ are stated in [10]. The proof is based on Beurling-Deny's criterion and Gagliardo-Nirenberg inequality.…”

“…Before introducing a weight function, we also recall two identities for partial energy functionals proved in [10].…”

“…In [10], diffusion phenomena for (1.1) with an exterior domain and for general radially symmetric damping term are obtained. However, the weighted energy estimates and diffusion phenomena for (1.1) with non-radially symmetric damping are still remaining open.…”

“…The difficulty seems to come from the choice of auxiliary function A in the weighted energy, which strongly depends on the existence of positive solution to the Poisson equation ∆A(x) = a(x). In fact, an example of non-existence of positive solution to ∆A = a for non-radial a(x) is shown in [10]. Radu, Todorova and Yordanov [9] considered the case Ω = R N and used a solution A * (x) of ∆A * = a 1 (1 + |x|) −α with a 1 > 0 satisfying…”

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.…”