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The solvability of the fractional partial differential equation with integral overdetermination condition for an inverse problem is investigated in this paper. We analyze the direct problem solution by using the "energy inequality" method. Using the fixed point technique, the existence and uniqueness of the solution of the inverse problem on the data are established. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate certain type superlinear nonlocal problems that are a class of parabolic equations with the second-type integral condition. We use the Fadeo-Galarkin method to establish the existence of the weak solution and we prove the uniqueness of this solution for the problem by using an a priori estimate. In addition, we study the theoretical blow-up solution and perform several numerical simulations of finite-time blow-up of a particular example of the main problem. [ABSTRACT FROM AUTHOR]