Existence of global solutions for a weakly coupled system of semilinear viscoelastic damped $\sigma$-evolution equations

Abstract

We prove the global (in time) existence of small data solutions from energy spaces based on $L^q$ spaces, $q\in \left(1,\infty \right)$, to the Cauchy problem for a weakly coupled system of semilinear viscoelastic damped $\sigma$-evolution equations, where we consider nonlinearity terms with powers $p_1,p_2>1$ and any ${\sigma }_1,{\sigma }_2\ge 1$ in the comparison between two single equations. To do this, by mixing additional $L^m$ regularity for the data on the basis of $L^q$-$L^q$ estimates, with $q\in \left(1,\infty \right)$ and $m\in \left[1,q\right)$, we apply $\left(L^m\cap L^q\right)$-$L^q$ 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 ${\sigma }_1$, ${\sigma }_2$, $m$ and $q$ bring some benefits to relax the restrictions on the admissible exponents $p_1,p_2$.