Existence and regularity results for space-time fractional integro-differential equation of Kirchhoff type with memory

This paper analyses a Kirchhoff type quasilinear space-time fractional integro-differential equation with memory $({\mathcal{K}}_{\alpha }^s)$. Various a priori bounds are derived in different norms on the solution of the considered equation. Utilizing these a priori bounds, existence and uniqueness of the weak solution to the proposed model are proved. Furthermore, regularity results on the solution of $({\mathcal{K}}_{\alpha }^s)$ are established. The contribution made in this work provides a framework for further investigation of such types of partial integro-differential equation $({\mathcal{K}}_{\alpha }^s)$.