On Nonexistence of Global Solutions for a Class of Fractional Integro-differential Problems

We propose to investigate the nonexistence of solutions of a general class of fractional integro-differential inequalities that consist of three terms: a derivative of first order, a fractional derivative and a nonlinear source term. We will consider the Riemann-Liouville and Caputo fractional derivatives of sub-first and sub-second orders. The nonlinear source term consists of a convolution of a (possibly singular) kernel with a polynomial of the state. Our objective is to establish various criteria under which there are no (nontrivial) global solutions. For this purpose, we plan to apply the test function method to the weak formulation of the problem. Then, we use suitable estimation techniques, asymptotic theories, and inequalities to obtain our results. We will seek some nontrivial examples to illustrate our findings. Our results could be utilized to identify the limitations of many physical systems and to analyze the behaviour of the solutions of some nonlinear fractional integro-differential equations and inequalities for which the explicit solution many not be available. Also our results will extend the abundant works for integer order problems to the limited results available for fractional order problems.