No formation of a new phase for a free boundary problem in combustion theory

We consider a free boundary problem for the heat equation with a given non-negative external heat source, which is often called a parabolic Bernoulli free boundary problem. 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.