2023/05/09 by Everaldo M. Bonotto, Bonotto, Everaldo M., Alexandre N. Carvalho +5
Engineering · Mathematics · #35B41 #35K40 (Secondary) #37B55 (Primary) 35B40 #Advanced Mathematical Physics Problems #Dynamical Systems (math.DS) #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2305.05724
openalex publication_date 2023/05/09 · openalex created_date 2023/05/12 · openalex updated_date 2026/07/28
The aim of this paper is to study the robustness of the family of pullback attractors associated to a non-autonomous coupled system of strongly damped wave equations, given by the following evolution system \ utt - Δu + u + η(-Δ)1/2ut + aε(t)(-Δ)1/2vt = f(u), · amp;(x, t) ∈Ω× (τ, ∞),
vtt - Δv + η(-Δ)1/2vt - aε(t)(-Δ)1/2ut = 0, · amp;(x, t) ∈Ω× (τ, ∞),. subject to boundary conditions u = v = 0, (x, t) ∈∂Ω× (τ, ∞), and initial conditions u(τ, x) = u0(x), ut(τ, x) = u1(x), v(τ, x) = v0(x), vt(τ, x) = v1(x), x ∈ Ω, τ∈ℝ, where Ω is a bounded smooth domain in ℝn, n ≥ 3, with the boundary ∂Ω assumed to be regular enough, η> 0 is a constant, aε is a Hölder continuous function satisfying uniform boundedness conditions, and f∈ C1(ℝ) is a dissipative nonlinearity with subcritical growth. This problem is a modified version of the well known Klein-Gordon-Zakharov system. Under suitable hyperbolicity conditions, we obtain the gradient-like structure of the limit pullback attractor associated with this evolution system, and we prove the continuity of the family of pullback attractors at ε= 0.