Direct and Inverse Initial Boundary Value Problems for Heat Equation with Non-Classical Boundary Condition

In this paper we consider solvability of an initial boundary value problem for the heat equation with a dynamic type boundary condition. Using some regularity and consistency conditions, the existence, uniqueness and continuous dependence upon the data of the classical solution are shown. This paper also considers an inverse problem of finding a time-dependent coefficient of the heat equation from the data of integral overdetermination condition. Conditions for the well-posedness of the formulated problem are found. In contrast to previous works, in this paper the existence and uniqueness of the solution of direct and inverse problems is proved without using orthogonality conditions.