Stability and continuous dependence of solutions of one-phase Stefan problems for semilinear parabolic equations (Q701341)

Consider the following reaction-diffusion problem with free boundary: \[ \begin{aligned} u_t-u_{xx}&= u^p, \quad 0<t<T, \;0<x<s(t),\\ u(0,x)&=u_0(x), \quad 0<x<s_0,\;s(0)=s_0,\\ u_x(t,0)&=0,\quad 0<t<T,\\ u(t,s(t))&=0, \quad s'(t)=-u_x(t,s(t)), \;0<t<T,\end{aligned} \] where \(p>1\) is a fixed real number. The author establishes the following results: 1) Stability of global fast solutions, i.e., if \((u,s)\) is global solution such that \(\lim_{t\to \infty}s(t)<\infty\) and \(|u(t)|_\infty \leq Ce^{-\alpha t}\), \(t\geq 0\) for some numbers \(C,\alpha>0\) and if the initial data \((\overline u_0, \overline s_0)\) are sufficiently close to \(( u_0, s_0)\), then \((\overline u, \overline s)\) is a global solution with the same properties when \(t\to \infty\). 2) Continuous dependence of local solutions up to the maximum existence time.