Global existence of solutions for a weakly coupled system of three damped ?-evolution equations

Abstract

In this paper our purpose is the study of the Cauchy problem for weakly coupled system of three semi-linear damped ?-evolution equations. Using (L? ? L²) -L² linear estimates combined with fractional Gagliardo-Nirenberg inequality. We find the so-called (p1 ´ p2 ´ p3) planes for the global (in time) existence. Moreover, from the interaction between the parameters m1, m2, m3 ? [1, 2) in one hand and ?1, ?2, ?3 ? 1 in the other hand. We proved lower bounds for powers nonlinearities similarly to the modified Fujita exponent, which are in the form of planes (p1´p2), (p1´p3) and (p2´p3).