Global Small Data Solutions for an Evolution Equation with Structural Damping and Hartree-Type Nonlinearity

In this paper, we consider an evolution equation with structural damping and nonlocal nonlinearity of Hartree type, and we prove the existence of global small data solutions for supercritical powers

Self-similar Asymptotic Profile for a Damped Evolution Equation

In this note, we describe the self-similar asymptotic profile for evolution equations with strong, effective damping.We assume that initial data are in weighted Sobolev spaces, and the second data verifies suitable moment conditions. The asymptotic profile is obtained by means of the application of a differential operator given by a linear combination of Riesz potentials to the fundamental solution of a (polyharmonic) diffusive problem

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

A wave equation with structural damping and nonlinear memory

In this paper, we obtain the critical exponent for a wave equation with structural damping and nonlinear memory: (Formula presented.) where μ > 0. In the supercritical case, we prove the existence of small data global solutions, whereas, in the subcritical case, we prove the nonexistence of global solutions for suitable arbitrarily small data, in the special case μ = 2