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

Cannon, John R.; van der Hoek, John

Boll. Unione Mat. Ital., Suppl. 1981, No. 1, 253-282 (1981).

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

MSC:

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)

Keywords:

integral equation method; existence; uniqueness; continuous dependence; potential theoretic methods