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

Abstract

In this paper we study the asymptotic profile (as t → ∞) of the solution to the Cauchy problem for the linear plate equation u t t + Δ 2 u − λ ( t ) Δ u + u t = 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 u t t + Δ 2 u − λ ( t ) Δ u + u t = | u | p . In order to do that, we assume small data in the energy space and, possibly, in L 1 . In this latter case, we also determinate the asymptotic profile of the solution to the semilinear problem for supercritical power nonlinearities.