Asymptotic profiles and critical exponents for a semilinear damped plate equation with time-dependent coefficients

Article type: Research Article

Authors: D’Abbicco, Marcello^{a; *} | Ebert, Marcelo Rempel^b

Affiliations: [a] Department of Mathematics, University of Bari, Via E. Orabona 4 70125, Bari, Italy. E-mail: [email protected] | [b] Departamento de Computação e Matemática, University of São Paulo, Ribeirão Preto, SP, 14040-901, Brazil. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: In this paper we study the asymptotic profile (as t→∞) of the solution to the Cauchy problem for the linear plate equation utt+Δ2u−λ(t)Δu+ut=0 when λ=λ(t) is a decreasing function, assuming initial data in the energy space and verifying a moment condition. For sufficiently small data, we find the critical exponent for global solutions to the corresponding problem with power nonlinearity utt+Δ2u−λ(t)Δu+ut=|u|p. In order to do that, we assume small data in the energy space and, possibly, in L1. In this latter case, we also determinate the asymptotic profile of the solution to the semilinear problem for supercritical power nonlinearities.

Keywords: Asymptotic profile, diffusion phenomena, critical exponents, plate equation

DOI: 10.3233/ASY-201624

Journal: Asymptotic Analysis, vol. 123, no. 1-2, pp. 1-40, 2021