Dynamics of thermally-insulated nonequilibrium Stefan problem (Q2644651)

The paper deals with two-phase Stefan problem with kinetics. Mathematically the problem can be posted as: find \(s(t)\) and \(u(x,t)\) such that \[ \begin{aligned} &u_t = u_{xx}, \quad x \not = s(t), \quad t>0,\\ &u(x,0) = u_0(x) \geq 0,\\ &g\left [ u(s(t), t) \right ] = \nu(t) < 0,\\ &\left [ u_x(s(t),t) \right ] := u_x^{+}(s(t),t) - u_x^{-}(s(t),t) = \nu(t).\end{aligned}\tag{1} \] In the above model, \(g\) represents the kinetics, \(\nu(t)\) is the interface velocity, \(s(t) = \int_0^t \nu(\tau)d\tau\) is its position and \(u\) is the temperature. Under some natural conditions on \(g\) the authors show that if the initial data \(u_0\) lies in a weighted Sobolev space \(H_\omega\), then {\parindent=8mm \begin{itemize}\item[(i)] There exists one and only one solution to the free interface problem (1) such that \(u \in C([0,\infty), H_w) \cap C((0,\infty), H_w^1)\) and \(\nu \in C(0,\infty)\). \item[(ii)] Moreover, \(\| u(\cdot,t)\| _{\omega} \leq C\) for all \(t\) and \(\| u(\cdot,t)\| _{{\omega,1}} \leq C_1(t_0)\) for \(t\geq t_0 >0\). \item[(iii)] The solution is classical and uniformly bounded for all \(t \geq t_0 > 0\). \end{itemize}} The authors also study existence of absorbing set and finite dimensional attractor for the problem without heat losses.