A well-posed problem for the heat equation
Thomas I. Seidman · Bulletin of the American Mathematical Society · 1974
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 equationIt is well known that arbitrary specification of both the initial state w 0 =«(0, •) and either Dirichlet data:or Neumann data:determines uniquely the evolution of the process.In particular, the terminal state u T = u(T, •) is determined by either of the pairs (u 0 ,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 u T .THEOREM.The observation/prediction problem for the heat equation is well posed for any bounded region £2 in R n with smooth boundary d£l.I.e., in the above notation, the map:(f, g)\-*u T is well defined and continuous, using appropriate