Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile (Q6541038)

There is studied the one-phase one-dimensional Stefan problem with vanishing as \(t\to \infty\) given function in the Neumann condition on the fixed boundary with the unknown functions \(u(x,t)\) and \(s(t)\) \N\[ \Nu_t - u_{xx} =0, \ t > 0, \ 0 < x < s(t),\N\]\N\[\N-u_x(0,t) = \frac{h}{\sqrt{1+t}}, \ u(s(t),t) = 0, \ \dot{s}(t) = -u_x(s(t),t), \ t > 0,\tag{1}\N\]\N\[\Nu(x,0) = u_0(x), \ 0 < x < s(0) = b_0,\N\]\Nwhere \(h > 0\).\N\NAfter the change of variables \(\eta = \frac{x}{\sqrt{1+t}}, \ \tau = \ln(t+1) \), the problem (1) is reduced to another problem, where is determined the pair \(\big(U(\eta), \, \omega\big)\) as the self-similar solution of the stationary problem, where \(\omega\) is definite positive value depending on \(h\).\N\NUnder an assumption that \(\big(u,\, s\big) \) is the solution of the problem (1) with \(0\le u_0 \in W^{\,1,\,\infty}(0,+\infty), \ 0 < u_0(0), \ u_0(x) = 0\) \ in \([b_0\,, \infty)\) the authors prove \N\[\N\lim_{t\to\infty}\frac{s(t)}{\sqrt{1+t}} = \omega, \ \lim_{t\to\infty}\sup_{x/\sqrt{1+t}\,\in [0,\omega]}\big| u(x,t) - U(\frac{x}{\sqrt{1+t}})\big| = 0.\N\]