Weakly coupled system of semilinear structural <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg" display="inline" id="d1e23"><mml:mi>σ</mml:mi></mml:math>-evolution models with <mml:math xmlns:mml="http://www.w3.org/1998/Math

This paper focuses on the study of global existence (in time) of solutions to a weakly coupled system of Cauchy problem for semilinear σk-evolution models with δk-visco-elastic damping. The system consists of two equations, one involving the function u and the other involving the function v. The equations are characterized by a classical power nonlinearity and a derivative-type nonlinearity. The main objective is to investigate the relationship between the regularity assumptions on the initial data and the range of permissible exponents p1 and p2 in the power nonlinearity. The paper considers the system in a spatial domain Rn and a time domain (0,∞), with specific conditions on the parameters σ1, σ2, δ1, and δ2, under the symmetry property as well as the exponents p1 and p2. The initial data (u1,v1) are required to satisfy certain conditions in terms of their integrability and Sobolev regularity.