On the nonexistence of global solutions for a class of fractional integro-differential problems
We study the nonexistence of (nontrivial) global solutions for a class of fractional integro-differential problems in an appropriate underlying space. Integral conditions on the kernel, and for some degrees of the involved parameters, ensuring the nonexistence of global solutions are determined.
Unlike the existing results, the source term considered is, in general, a convolution and therefore nonlocal in time. The class of problems we consider includes problems with sources that are polynomials and fractional integrals of polynomials in the state as special cases.
Singular kernels illustrating interesting cases in applications are provided and discussed. Our results are obtained by considering a weak formulation of the problem with an appropriate test function and several appropriate estimations.