Global Existence of Small Data Solutions to Weakly Coupled Systems of Semi-Linear Fractional σ-Evolution Equations with Mass and Different Nonlinear Memory terms

We study in this paper the long-term existence of solutions to the system of weakly coupled equations with fractional evolution and various nonlinearities. Our objective is to determine the connection between the regularity assumptions on the initial data, the memory terms, and the permissible range of exponents in a specific equation. Using Lp-Lq estimates for solutions to the corresponding linear fractional sigma-evolution equations with vanishing right-hand sides, and applying a fixed-point argument, the existence of small data solutions is established for some admissible range of powers (p1,p2,& mldr;,pk).