Abstract
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 ‐evolution equations with friction and visco‐elastic type damping, where for . We study model (1.1) (see below) in several cases with respect to the regularity for the data: First, we assume data . By using estimates of Sobolev solutions to related linear models with vanishing right‐hand side, we explain the admissible range of exponents in (1.1) which allow to prove the global (in time) existence of small data Sobolev solutions. Then we suppose that the data belong to , where now , . So the data have additional regularity to the first assumed integrability. We restrict ourselves to data from energy space (on the basis of ) and from energy space with additional suitable higher regularity. We compare in our statements the admissible set of exponents and in the power nonlinearities with the so‐called modified Fujita exponent. Finally, some blow‐up results complete the paper.
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