Accurate asymptotics for singularly perturbed dynamic free boundary problems (Q1174848)

The author considers the behaviour of a sufficiently regular solution \(u\) of the free boundary problem \[ \partial u/\partial t-\partial^ 2u/\partial x^ 2+u-f\leq 0,\quad u\leq 0\quad\text{ a.e. in }(0,1)\times(0,T), \] \[ \left(\partial u/\partial t-\partial^ 2u/\partial x^ 2+u-f\right) u=0, \] \[ u(x,0)=\overline u(x)\quad\text{ for }\quad x\in(0,1) \quad\text{ and }\quad u=0 \quad\text{ on }\quad \{0,1\}\times(0,T). \] If \(\varepsilon>0\) is a small parameter, the problem is of a singularly perturbed type, in the sense that \(\varepsilon\) multiplies the highest order derivative in the equation. The analysis of the problem contains two main elements: (i) a discussion of the structure of a formal approximation of the solution and the free boundary for \(\varepsilon\searrow 0\) and (ii) concrete error estimates in the maximum norm showing the correctness of the highest order term of the formal approximation.