Global classical solution of quasi-stationary Stefan free boundary problem
The quasi-stationary Stefan problem (i.e. the Stefan problem with zero specific heat) is also known as the Hele-Shaw problem. The author proves its classical solvability, also discussing the long time behaviour of the solutions. For simplicity of exposition the paper refers to the one-phase, two-dimensional problem, but the results are true in general. The only restriction adopted is that the initial and boundary data are small enough perturbation of the data of a one-dimensional problem. The problem to be solved is the following: find the pair \((u(x,y,t),\rho (x,t))\) satisfying \[ \begin{aligned} \Delta u=0,\quad & x\in\mathbb R,\quad 0<y<\rho(x,t),\quad t>0,\\ u(x,y,t)=0,\quad & x\in\mathbb R,\quad y=\rho(x,t),\quad t>0,\\ V_n=-\dfrac{\partial u}{\partial n},\quad & x\in\mathbb R,\quad y=\rho(x,t),\quad t>0\\ u(x,0,t)=g(x,t),\quad & x\in\mathbb R,\quad t>0,\\ \rho(x,0)=\rho_0(x),\quad & x\in\mathbb R,\end{aligned} \] and the solution is looked for in the form \[ u(x,y,t)=u_0(y,t)+\epsilon u_1(x,y,t), \] \[ \rho(x,t)=s(t)+\epsilon\sigma (x,t). \] Perturbations \(u_1, \sigma\) satisfy the system \[ \begin{aligned} \Delta u_1=0,\quad & x\in\mathbb R,\quad 0<y<\rho(x,t),\quad t>0,\\ u_1=\dfrac{g(t)}{s(t)}\sigma(x,t),\quad & x\in\mathbb R,\quad y=\rho(x,t),\quad t>0,\\ \partial_t\sigma+\partial_y u_1-\epsilon\partial_x\sigma\partial_x u_1=0,\quad & x\in\mathbb R,\quad y=\rho(x,t),\quad t>0,\\ u_1(x,0,t)=g_1(x,t),\quad & x\in\mathbb R,\quad t>0,\\ \sigma(x,0)=\sigma_0(x),\quad & x\in\mathbb R \end{aligned} \] which is shown to have a classical solution provided \(\epsilon\) is small enough. A problem more general than the latter is in fact studied. A fixed point technique is used, going through the determination of delicate estimates. Finally it is shown that when the perturbations of the data belong to a suitable class, then \(u_1,\sigma\) decay asymptotically at least as \((1+t)^{-1}, (1+t)^{-\frac{1}{2}}\), respectively.
scientific article; zbMATH DE number 2136867