On the asymptotic behavior of the energy for evolution models with oscillating time-dependent damping

Article type: Research Article

Authors: Aslan, Halit Sevki | Rempel Ebert, Marcelo^{; *}

Affiliations: Department of Computer Science and Mathematics (FFCLRP), University of São Paulo (USP), Ribeirão Preto, SP, 14040-901, Brazil

Correspondence: [*] Corresponding author. E-mail: ebert@ffclrp.usp.br.

Abstract: In the present paper, we study the influence of oscillations of the time-dependent damping term b(t)ut on the asymptotic behavior of the energy for solutions to the Cauchy problem for a σ-evolution equation utt+(−Δ)σu+b(t)ut=0,(t,x)∈[0,∞)×Rn,u(0,x)=u0(x),ut(0,x)=u1(x),x∈Rn, where σ>0 and b is a continuous and positive function. Mainly we consider damping terms that are perturbations of the scale invariant case b(t)=β(1+t)−1, with β>0, and we discuss the influence of oscillations of b on the energy estimates according to the size of β.

Keywords: Damped σ-evolution equation, time-dependent damping, energy estimates, effective dissipation, non-effective dissipation

DOI: 10.3233/ASY-231851

Journal: Asymptotic Analysis, vol. 135, no. 1-2, pp. 185-207, 2023