The existence of and a continuous dependence result for the solution of the heat equation subject to the specification of energy. (English) Zbl 0538.35038

The author shows existence, uniqueness, and continuous dependence on the domain for the solution of the heat equation subject to a boundary condition of the form \(E(t)=\int^{s(t)}_{0}u(x,t)dt,\) which can be understood as a specification of the energy content of material at each time t. The author uses mainly potential theoretic methods. The results are deduced from a somehow cumbersome analysis of the solutions to a system of integral equations which arise from representing the solution in terms of potentials. Main tools are: 1) the use of a suitable norm, 2) the calculation of preliminary estimates for the integral operators involved, 3) the use of contraction arguments.
Reviewer: W.Manntz


35K05 Heat equation
35C15 Integral representations of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B45 A priori estimates in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)