Parabolic problems of free boundary (Q2717326)

This is a paper on the following class of free boundary problems \(u_t=\nabla u+f(u, Du)\), \(t\geq 0\), \(q \in \Omega_t\) \(u(t, \eta)=g_1(\eta)\), \(t\geq 0\), \(\eta \in \partial \Omega_t,\) \(\frac{\partial}{\partial \nu}u(t, \eta)=g_2(\eta)\), \(t\geq 0\), \(\eta \in \partial \Omega_t,\) complemented with initial conditions for the unknown domain \(\Omega_t\) and for \(u\). The space variable \(\eta\) varies in \(\mathbb{R}^n\). After emphasizing the substantial difference between the case \(n=1\) and the case \(n>1\), the author illustrates how to reduce the problem to a problem in a fixed domain for an equation which is nonlocal and completely nonlinear. This is done first for one-phase and then for two-phase problems. Next the main results concerning existence and uniqueness are illustrated together with the techniques generally employed. Finally, applications to specific cases (with particular reference to combustion theory) are discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00026].