2024/08/01 by Thinh Dang, Bacim Alali, Dang, Thinh +3 · 2 citations
Computer Science · Engineering · Mathematics · #45A05 #45M15 #45P05 #47G10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2408.00912
openalex publication_date 2024/08/01 · openalex created_date 2024/08/29 · openalex updated_date 2026/07/28
This work addresses the regularity of solutions for a nonlocal wave equation over the space of periodic distributions. The spatial operator for the nonlocal wave equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular as well as distributional solutions of the nonlocal wave equation, integrable as well as singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kernels to nonlocal-pseudo differential operators. We present results on the spatial and temporal regularity of solutions in terms of regularity of the initial data or the forcing term. Moreover, solutions of the nonlocal wave equation are shown to converge to the solution of the classical wave equation for two types of limits: as the spatial nonlocality vanishes or as the singularity of the integral kernel approaches a certain critical singularity that depends on the spatial dimension.