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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 L p − L q estimates for solutions to the corresponding linear fractional σ –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 (p 1 , p 2 , ... , p k) . [ABSTRACT FROM AUTHOR]

In this paper, we consider a weakly system of fractional θ -evolution equations. Using the fixed-point theorem, a global-in-time existence of small data solutions to the Cauchy problem is proved for one single equation. Using these results, we prove the global existence for the system under some mixed symmetrical conditions that describe the interaction between the equations of the system. [ABSTRACT FROM AUTHOR]

We prove the global existence of small data solution in all spaces of all dimensions n ≥ 1 for weakly coupled systems of semilinear effectively damped wave, with different time-dependent coefficients in the dissipation terms. Moreover, we assume that the nonlinearity terms f (t , u) and g (t , v) satisfy some properties of parabolic equations. We study the problem in several classes of regularity. [ABSTRACT FROM AUTHOR]