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 $(\mathcal{K}^{s}_{\alpha})$. 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 $(\mathcal{K}^{s}_{\alpha})$ are established. The contribution made in this work provides a framework for further investigation of such types of partial integro-differential equation $(\mathcal{K}^{s}_{\alpha})$.