On Nonexistence of Global Solutions for a System of Strongly Coupled Fractional Differential Equations

The study of the nonexistence of solutions for systems of differential equations is as important as the study of existence of solutions. It is particularly capital for the nonlinear differential equations and systems where solutions cannot be found explicitly. In this project we propose to investigate the nonexistence of non-trivial global solutions for a system of two strongly coupled fractional differential equations. Each equation involves two fractional derivatives and a nonlinear source term. The fractional derivatives are of Caputo type of sub-first orders. The nonlinear sources take the form of a convolution of a (possibly singular) kernel with a polynomial of the state. We intend to establish some criteria under which there are no (nontrivial) global solutions. For this purpose, we plan to work on the weak formulation of the system and select an appropriate test function. Using some suitable estimations and inequalities we should be able to arrive at a contradiction in case we assume the contrary. Some examples will be provided to illustrate the results. Our findings can be used to characterize the restriction of many physical systems and to analyze the behavior of solutions for the nonlinear fractional systems for which explicit solutions may not be computed. In the area of mathematical analysis, our results will extend the results for integer order systems to the fractional-order systems.