A structurally damped σ-evolution equation with nonlinear memory

In this paper, we investigate the global (in time) existence of small data solutions to the Cauchy problem for the following structurally damped sigma-evolution model with nonlinear memory term: u(tt)+(-Delta)(sigma)u+mu(-Delta)(sigma/2)u(t) = integral(t)(0) (t-tau)(-gamma)|u(t)(tau, center dot)|(p) d tau, with sigma>0. In particular, for gamma is an element of((n-sigma)/n,1), we find the sharp critical exponent, under the assumption of small data in L-1. Dropping the L-1 smallness assumption of initial data, we show how the critical exponent is consequently modified for the problem. In particular, we obtain a new interplay between the fractional order of integration 1-gamma in the nonlinear memory term and the assumption that initial data are small in L-m, for some m>1.