The free boundary problem without initial condition (Q1948551)

The paper under review investigates the problem of determination of a pair of functions \((s(t), u(t,x))\) of which \(u(t,x)\) satisfies the heat equation \[ u_{xx}(t,x) = u_{t}(t,x), \;(t,x)\in D_T, \] with the boundary conditions \[ u_x(t,0) = f(t, u(t,0)), \] \[ u_x(t,s(t)) = 0,\;0\leq t \leq T, \] \[ u(t,s(t)) = \int\limits_{0}^t q(\tau,s(\tau)) d\tau, \;0 \leq t\leq T. \] Here the function \(s(t)\) is continuously differentiable on \([0,T]\), \(s(0) = 0\), \(s(t) > 0\), and \(D_T = \{(t,x): 0\leq t\leq T, 0 < x < s(t)\}\). The function \(s(t)\) represents the free boundary of the domain \(D_T\). Under some assumptions on the monotonicity of the given functions \(f(t, x)\) and \(q\), the authors proved the existence and uniqueness of a solution to the above problem. They also investigated the asymptotic behavior as \(t\to \infty\) of the boundary function \(s(t)\).