A well-posed problem for the heat equation

Abstract

Simultaneous specification of (consistent) Dirichlet and Neumann data boundedly determines later internal states of the solution of the heat equation in a general region. We consider solutions of the heat equation ut=au for 0 < / < r , x= (xl9 • • • , xn) e Q. It is well known that arbitrary specification of both the initial state w0=«(0, •) and either Dirichlet data: u(t, x) = f(t, x) for 0 <, / <; T, x e 9Q, or Neumann data: (du/dv)(t, x) = g(t, x) for 0 <; t <± T, x e dQ, determines uniquely the evolution of the process. In particular, the terminal state uT = u(T, •) is determined by either of the pairs (u0,f), If the initial internal state is not given, we ask whether knowledge of both Dirichlet and Neumann data suffices. The pair (ƒ, g) cannot be specified arbitrarily, but we adopt the viewpoint that in observation of an ongoing process, the consistency conditions are automatically satisfied so the observed pair (ƒ, g) lies in the admissible manifold M, and the existence of a solution is not at issue. We ask whether observation of the boundary data (ƒ, g) suffices for effective prediction of the terminal internal state uT. THEOREM. The observation/prediction problem for the heat equation is well posed for any bounded region £2 in R with smooth boundary d£l. I.e., in the above notation, the map:(f, g)\-*uT is well defined and continuous, using appropriate <JS?2 topologies for domain and range. SKETCH OF PROOF, (a) There is a reduction to a restricted problem of the same form with ^ = 0 . AMS (MOS) subject classifications (1970). Primary 35K05, 93B05.