Study of a Superlinear Problem for a Time Fractional Parabolic Equation Under Integral Over-determination Condition

The main purpose of this paper is to examine the inverse problem associated with determining the right-hand side of a nonlinear fractional parabolic equation. This equation is accompanied by an integral over-determination supplementary condition. With the use of the functional analysis method, we establish the continuity, existence and uniqueness based on the construction of the direct problem. Such a method relies on the density of the range of the operator established for the problem at hand coupled with the energy inequality scheme. This scheme, also referred to as the method of a priori estimates, allows us to derive the existence theorem from the solution of the given problem, starting with the uniqueness theorem. For the solvability of the inverse problem and its uniqueness, we establish certain suitable conditions, and to demonstrate the existence and uniqueness of its solution, we utilize the fixed point theorem.