A finite-dimensional attractor for a nonequilibrium Stefan problem with heat loss (Q556475)

The following one dimensional, two-phase Stefan problem is considered \[ u_t=u_{xx}-\gamma u,\,\,\,x \neq s(t),\,\,\,t>0, \] \[ u(x,0)=u_0(x)\geq0, \] \[ g[u(s(t),t)]=\dot{s}(t), \] \[ [u_x(s(t),t)]:=u^+_x(s(t),t)-u^-_x(s(t),t)=v(t). \] The coefficient \(\gamma\) is non-negative and is associated to heat loss. The function \(g\) is continuously differentiable, \(g'<0\), and sublinear. The main results are concerned with the asymptotic behaviour of the solution. The existence of a compact attractor is proved, whose Hausdorff dimension is estimated from above when \(\gamma>0\).