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

In this paper we study the asymptotic profile (as~$t oinfty$) of the solution to the Cauchy problem for the linear plate equation % [ u_{tt}+Delta^2 u - lambda(t)Delta u + u_t =0 ] % when~$lambda=lambda(t)$ 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_{tt}+Delta^2 u - lambda(t)Delta 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.