Global existence for semi-linear structurally damped σ-evolution models

The main purpose of this paper is to study the global existence of small data solutions for semi-linear structurally damped a-evolution models of the form u(tt) + (-Delta)(sigma)u + u(-Delta)(delta)(ut) = f (broken vertical bar D broken vertical bar(a) u, ut), u(0, x) = u(o) (x), ut (0, x) = u1(x) with sigma >= 1, mu > 0 and delta = sigma/2. This is a family of structurally damped or-evolution models interpolating between models with exterior damping delta = 0 and those with visco-elastic type damping delta = sigma. The function f(broken vertical bar D broken vertical bar(a), ut) represents power non-linearities parallel to D parallel to(a)u broken vertical bar(p) for a is an element of [0, sigma) or broken vertical bar ut broken vertical bar(p). Our goal is to propose a Fujita type exponent diving the admissible range of powers p into those allowing global existence of small data solutions (stability of zero solution) and those producing a blow-up behavior even for small data. On the one hand we use new results from harmonic analysis for fractional Gagliardo Nirenberg inequality or for superposition operators (see Appendix A), on the other hand our approach bases on L-p - L-q estimates not necessarily on the conjugate line for solutions to the corresponding linear models assuming additional L-1 regularity for the data. The linear models we have here in mind are u(tt)(-Delta)sigma upsilon + mu(-Delta)delta(upsilon t) = 0, upsilon(0, x) = upsilon(0) (x), upsilon(t) (0, x) = upsilon i(x) with sigma >= 1, mu > 0 and delta is an element of (0, sigma]. (C) 2015 Elsevier Inc. All rights reserved.