Fast energy decay for 2-D wave equation with localized damping near spatial infinity

2025/09/17 by Ikehata, Ryo
#35B45 #35L05 #35L15 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.13645

Abstract

We consider the Cauchy problem for wave equations with localized damping in \bf R². The damping is effective only near spatial infinity. We obtain fast energy decay estimate such that O(t^-2log t) as t → ∞. Unlike the results for the two-dimensional exterior mixed problem case, the difficulty of not being able to use Hardy-type inequalities is overcome by using Poincaré-type inequalities in all spaces and the finite propagation property of the solution to construct an estimate formula. In the two-dimensional case, when comparing the problem in the whole space with that in the exterior domain, we find that there is a significant difference in the sense that the former requires a logarithmic correction to the energy decay rate.

Citations

• On the system of 2-D elastic waves with critical space dependent damping
• L²-blowup estimates of the wave equation and its application to local energy decay
• Energy decay and diffusion phenomenon for the asymptotically periodic damped wave equation
• Energy decay for linear dissipative wave equations in exterior domains

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