Weakly coupled systems of semilinear σ-evolution equations with friction and visco-elastic type damping

In the present paper, we are interested to study global (in time) existence of small data Sobolev solutions to the Cauchy problem for a weakly coupled system of two semilinear sigma(k)-evolution equations with friction and visco-elastic type damping, where sigma(k) >= 1 for k = 1, 2. We study model (1.1) (see below) in several cases with respect to the regularity for the data: First, we assume data (u(1), v(1)) is an element of (L-1(m) x L-2(m)) . By using L-m - L-q estimates of Sobolev solutions to related linear models with vanishing right-hand side, we explain the admissible range of exponents (p(1), p(2)) in (1.1) which allow to prove the global (in time) existence of small data Sobolev solutions. Then we suppose that the data (u(1), v(1)) belong to (L-1(m) boolean AND H-q(1)s) x ( L-2(m) boolean AND H-q(2)s), where now m(k) is an element of (1, q), is an element of (1, infinity). So the data have q additional regularity to the first assumed L-1(m) x L-2(m) integrability. We restrict ourselves to data from energy space (on the basis of L-q) and from energy space with additional suitable higher regularity. We compare in our statements the admissible set of exponents p(1) and p(2) in the power nonlinearities with the so-called modified Fujita exponent. Finally, some blow-up results complete the paper.