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.

A structurally damped \sigma ‐evolution equation with nonlinear memory / D'Abbicco, Marcello; Girardi, Giovanni. - In: MATHEMATICAL METHODS IN THE APPLIED SCIENCES. - ISSN 0170-4214. - (2020). [10.1002/mma.6633]

A structurally damped \sigma ‐evolution equation with nonlinear memory

Giovanni Girardi
2020-01-01

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/314981
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