An application of L^m-L^r estimates to weakly coupled systems of semilinear viscoelastic wave equations

2018/09/20 by Yan Liu, Wenhui Chen, Liu, Yan +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1809.07525

openalex publication_date 2018/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider weakly coupled systems of semilinear viscoelastic wave equations with different power source nonlinearities in ℝⁿ, n≥1 as follows: \\beginaligned amp;u_tt-Δu+g∗Δu+u_t=|∂_t^ℓv|^p,
amp;v_tt-Δv+g∗Δv+v_t=|∂_t^ℓu|^q,
\endaligned. with ℓ=0,1 and p,q>1. After presenting L^m-L^r estimates with 1≤ m≤ r≤ ∞ of solutions to the corresponding linearized problem with vanishing right-hand side, we prove the existence of global in time solutions to the weakly coupled systems, where the initial data are supposed to belong to different L^r spaces with different additional L^m regularities.

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