No formulation of a new phase for a free boundary problem in combustion theory
Ken Furukawa, Yoshikazu Giga, Naoto Kajiwara
TL;DR
The paper investigates whether a new unburnt phase can form in a free-boundary heat equation with a nonnegative external source under zero-temperature initial data. It employs a fixed-domain transformation and Laplace-transform techniques, combined with a maximum-principle argument, to prove nonexistence of nonnegative solutions when $u_0\equiv0$ and $f$ is bounded above, in both 1D and certain higher-dimensional geometries. It also analyzes a self-similar solution in the singular forcing regime $f(s,t)=h/\sqrt{t}$ for $N=1$, deriving a Hermite-type ODE and establishing existence of a unique scaling parameter leading to a self-similar profile. Collectively, the results establish a no-new-phase principle for bounded forcing and provide a constructive example under singular forcing, with implications for combustion theory and the behavior of the unburnt zone near a burning front.
Abstract
We consider a free boundary problem for the heat equation with a given non-negative external heat source. On the free boundary, we impose the zero Dirichlet condition and the fixed normal derivative so that heat escapes from the boundary. In various settings, we show that there exist no solutions when the initial temperature equals the fixed temperature no matter where the initial location of the free boundary is given provided that the external heat source is bounded from above. We also note that there is a chance to have a solution when the external temperature is unbounded as time tends to zero by giving a self-similar solution.