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 $u$ 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 $p_1$ and $p_2$ in the power nonlinearity. The paper considers the system in a spatial domain $R^n$ and a time domain $(0,\infty )$, 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.