Uniqueness of solution to a free boundary problem from combustion

We investigate the uniqueness and agreement between different kinds of solu tions for a free boundary problem in heat propagation that in classical ter ms is formulated as follows: to find a continuous function u(x, t) greater than or equal to 0, defined in a domain D subset of R-N x (0, T) and such t hat Deltau + Sigma a(i) u(xi) - u(t) = 0 in D boolean AND {u > 0}. We also assume that the interior boundary of the positivity set, D boolean AND partial derivative {u >0}, so-called free boundary, is a regular hypers urface on which the following conditions are satisfied: u = 0, -partial derivativeu/partial derivativev = C. Here v denotes outward unit spatial normal to the free boundary. In additio n, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of D. This problem arises in combustion theory as a limit situation in the propagation of premixed ames (high activation ene rgy limit). The problem admits classical solutions only for good data and for small tim es. Several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. We investigate cond itions under which the three concepts agree and produce a unique solution.