Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Abstract

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

Nonexistence of Global Solutions to Higher-Order Time-Fractional Evolution Inequalities with Subcritical Degeneracy

“…where I 1 and I 2 are given, respectively, by ( 17) and (18). Let ϕ be the test function given by (19).…”

“…Namely, under suitable conditions on the initial value u 0 , it was shown that if 1 < p < 1 + 2 N , then any nontrivial non-negative solution to (2)-( 6) (with k = 1) blows up in finite time, while if p ≥ 1 + 2 N and u 0 is sufficiently small with respect to a certain norm, then the problem admits global solutions. For other works related to nonexistence results for fractional evolution equations and inequalities, as can be seen in, e.g., [16][17][18][19] and the references therein. Motivated by the above contributions, our aim in this paper was to obtain sufficient conditions for which (1)-( 2) and ( 3)-( 2) have no global weak solutions in a sense which will be subsequently specified.…”