A structurally damped \sigma ‐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) vertical bar u(t)(tau, center dot)vertical bar(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.