2022/07/14 by Alessio Barbieri, Barbieri, Alessio, Enzo Vitillaro +1 · 1 citation
Computer Science · Mathematics · #35B35 #35C10 #35L05 #35L20 #35L51 #76Q05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2207.07047
openalex publication_date 2022/07/14 · openalex created_date 2022/07/16 · openalex updated_date 2026/07/28
The aim of the paper is to study the problem utt+dut-c2Δu=0 in ℝ×Ω, μvtt- divΓ(σ∇Γv)+δvt+κv+ρut =0 on ℝ× Γ1, vt =∂νu on ℝ× Γ1, ∂νu=0 on ℝ× Γ0, u(0,x)=u0(x), ut(0,x)=u1(x) in Ω, v(0,x)=v0(x), vt(0,x)=v1(x) on Γ1, where Ω is a open domain of ℝN with uniformly Cr boundary (N≥ 2, r≥ 1), Γ=∂Ω, (Γ0,Γ1) is a relatively open partition of Γ with Γ0 (but not Γ1) possibly empty. Here divΓ and ∇Γ denote the Riemannian divergence and gradient operators on Γ, ν is the outward normal to Ω, the coefficients μ,σ,δ, κ, ρ are suitably regular functions on Γ1 with ρ,σ and μ uniformly positive, d is a suitably regular function in Ω and c is a positive constant. In this paper we first study well-posedness in the natural energy space and give regularity results. Hence we study asymptotic stability for solutions when Ω is bounded, Γ1 is connected, r=2, ρ is constant and κ,δ,d≥ 0.