Nonexistence of global solutions for a fractional system of strongly coupled integro-differential equations

Abstract

In this paper, we study the nonexistence of nontrivial global solutions for a system of two strongly coupledfractional differential equations. Each equation involves two fractional derivatives and a nonlinear source term. Thefractional derivatives are of Caputo type of subfirst orders. The nonlinear sources are nonlocal in time. They have theform of a convolution of a polynomial of the state with a (possibly singular) kernel. The system under considerationis a generalization of many interesting special systems of equations whose solutions do not exist globally in time. Weestablish some criteria under which no nontrivial global solutions exist. Several integral inequalities and estimations arederived and the test function method is adopted. Special cases and examples are given to illustrate the results.