Uniqueness of solution to a free boundary problem from combustion

Abstract

We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function u ( x , t ) ≥ 0 , u(x,t)\geq 0, defined in a domain D ⊂ R N × ( 0 , T ) \mathcal {D} \subset {\mathbb {R}}^{N}\times (0,T) and such that \[ Δ u + ∑ a i u x i − u t = 0 in D ∩ { u > 0 } . \Delta u+\sum a_{i}\,u_{x_{i}}-u_{t}=0\quad \text {in}\quad \mathcal {D}\cap \{u>0\}. \] We also assume that the interior boundary of the positivity set, D ∩ ∂ { u > 0 } \mathcal {D} \cap \partial \{u> 0\} , so-called free boundary, is a regular hypersurface on which the following conditions are satisfied: \[ u = 0 , − ∂ u / ∂ ν = C . u=0 ,\quad -\partial u/\partial \nu = C. \] Here ν \nu denotes outward unit spatial normal to the free boundary. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of D \mathcal {D} . This problem arises in combustion theory as a limit situation in the propagation of premixed flames (high activation energy limit). The problem admits classical solutions only for good data and for small times. Several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. We investigate conditions under which the three concepts agree and produce a unique solution.