Local energy decay for 2-D wave equations with variable coefficients

2025/09/17 by Ryo Ikehata, Ikehata, Ryo
Engineering · Mathematics · #35B40 #35C20 #35L05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2509.13640

openalex publication_date 2025/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper addresses the two-dimensional initial value problem in \bf R² for the wave equation with varying spatial coefficients in the main part. Assuming compactness in the support of the initial value, we report that the corresponding local energy decays to an order of magnitude of, for example, O(t^-1√(log t)) after sufficiently large time. For the two-dimensional whole space case, it is crucial to establish the optimal L²-estimate for the solution itself, skillfully avoiding the difficulty of not being able to use useful inequalities such as Hardy-type inequalities in higher dimensional case. We also consider cases where the variable coefficients are slightly generalized. These proofs are developed using the multiplier method.

Citations

• On the system of 2-D elastic waves with critical space dependent damping
• Optimal L²-growth of the generalized Rosenau equation
• L²-blowup estimates of the wave equation and its application to local energy decay
• A Note on Decay Rates of the Local Energy for Wave Equations with Lipschitz Wavespeeds
• Local energy decay for Lipschitz wavespeeds

Related

• Fast energy decay for 2-D wave equation with localized damping near spatial infinity
• A note on local energy decay results for wave equations with a potential
• A Note on Decay Rates of the Local Energy for Wave Equations with Lipschitz Wavespeeds
• L²-blowup estimates of the wave equation and its application to local energy decay
• Special uniform decay rate of local energy for the wave equation with variable coefficients on an exterior domain