Weakly coupled system of semilinear structural σ-evolution models with δ- visco-elastic damping

This paper focuses on the study of global existence (in time) of solutions to a weakly coupled system of Cauchy problem for semilinear sigma(k)-evolution models with delta(k)-visco-elastic damping. The system consists of two equations, one involving the function v. 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 p(1) and p(2) in the power nonlinearity. The paper considers the system in a spatial domain R-n and a time domain (0, infinity), with specific conditions on the parameters sigma(1), sigma(2), delta(1), and delta(2), under the symmetry property as well as the exponents p(1) and p(2). The initial data (u(1), v(1)) are required to satisfy certain conditions in terms of their integrability and Sobolev regularity.