A free-boundary problem for the heat equation arising in flame propagation

Abstract

We introduce a new free-boundary problem for the heat equation, of interest in combustion theory. It is obtained in the description of laminar flames as an asymptotic limit for high activation energy. The problem asks for the determination of a domain in space-time, Ω ⊂ R n × ( 0 , T ) \Omega \subset {{\mathbf {R}}^n} \times (0,T) , and a function u ( x , t ) ⩾ 0 u(x,t) \geqslant 0 defined in Ω \Omega , such that u t = Δ u {u_t} = \Delta u in Ω , u \Omega ,\;u takes certain initial conditions, u ( x , 0 ) = u 0 ( x ) u(x,0) = {u_0}(x) for x ∈ Ω 0 = ∂ Ω ∩ { t = 0 } x \in {\Omega _0} = \partial \Omega \cap \{ t = 0\} , and two conditions are satisfied on the free boundary Γ = ∂ Ω ∩ { t > 0 } : u = 0 \Gamma = \partial \Omega \cap \{ t > 0\} :u = 0 and u ν = − 1 {u_u } = - 1 , where u ν {u_u } denotes the derivative of u u along the spatial exterior normal to Γ \Gamma . We approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data.