A qualitative study on a parabolic free boundary problem (Q1183074)

The problem \[ \begin{cases} u_{xx}=u_ t+f(t,x,u,u_ x) &\text{ for \(0<t\leq T, 0<x<s(t),\)} \\ (Bu)(t)\equiv au(t,0)-bu_ x(t,0)=\alpha(t) &\text{ in \((0,T] (a,b\geq 0, a+b=1)\),} \\ u(t,s(t))=u_ x(t,s(t))=0 &\text{ in \((0,T], u(0,0)=s(0)=0,\)}\end{cases}\tag{P} \] is considered. Some results: Nonexistence if \(f(t,x,0,0)\leq 0\); \(u_ x<0\) and hence \(u>0\) if \(f\) is (weakly) increasing in \(x\); \(u\) and \(s\) increasing in \(t\) if \(f\) is decreasing in \(t\) and \(\alpha\) increasing. The basic monotonicity theorem states roughly that, if in (P) \(f\) is made smaller and \(\alpha\) larger, then \(u\) and \(s\) become larger. Among the consequences are a uniqueness theorem, continuous dependence of \(u\) and \(s\) on \(f\) and \(\alpha\) and upper and lower bounds for \(u\) and \(s\).