Abstract

We prove the global (in time) existence of small data solutions from energy spaces based on Lq spaces, q∈(1,∞), to the Cauchy problem for a weakly coupled system of semilinear viscoelastic damped σ-evolution equations, where we consider nonlinearity terms with powers p1,p2>1 and any σ1,σ2≥1 in the comparison between two single equations. To do this, by mixing additional Lm regularity for the data on the basis of Lq-Lq estimates, with q∈(1,∞) and m∈[1,q), we apply (Lm∩Lq)-Lq estimates for solutions to the corresponding linear Cauchy problems to treat semilinear problems. In addition, two different strategies allowing no loss of decay and some loss of decay combined with the flexible choice of admissible parameters σ1, σ2, m and q bring some benefits to relax the restrictions on the admissible exponents p1,p2.