Asymptotic behavior of solutions to diffusion problems with Robin and free boundary conditions (Q2843421)

There is considered the boundary value problem with unknowns \(u(t,x)\) and free boundary \(h(t)\) NEWLINE\[NEWLINE\begin{cases} u_t=u_{xx}+ f(u),\, t>0,\, 0 < x < h(t),\\ u(t,0) = b u_x(t,0), \;t > 0,\\ u(t,h(t)) = 0, \;h'(t) = -\mu u_x(t,h(t)), \;t > 0,\\ h(0) = h_0, \;u(0,x) = u_0(x), \;0 \leq x \leq h_0, \end{cases}\tag{1}NEWLINE\]NEWLINE where \(\mu > 0\) and \(b \geq 0\) are constants, \(f\) satisfies: \(f \in C^1([0,\infty))\), \(f(0)=0\) and \(f(u)\leq Ku\) for some \(K > 0\). Let \(h_\infty :=\lim_{t\to\infty}h(t)\in (h_0, \infty ]\), \(I(t) := [0,h(t))\). The results of a present article are as follows. Let for all \(t>0\) \((u, h)\) be the solution of the problem (1) and \(0\leq u\leq C\), \(C>0\), then {\parindent=6mm\begin{itemize}\item[1.] \(u(t,\cdot)\) converges as \(t \to \infty\) locally uniformly with respect to \(x\) in \([0,\,h_\infty)\) to the solution of the problem NEWLINE\[NEWLINEv'' + f(v)=0,\, 0<x<h_\infty,\, v(0)=b v'(0); \tag{2}NEWLINE\]NEWLINE \item[2.] either \(h_\infty =\infty\) and \(\lim_{t\to\infty}u(t,x)=v^*(x)\), where \(v^*\) is a unique solution of (2) and \(0\leq v^*\leq C\) (spreading) or \(h_\infty\leq\frac{\pi}{\sqrt{f'(0)}} \) and \(\lim_{t\to\infty}\| u(t,\cdot)\|_{L^\infty(I(t))}=0\) (vanishing);\item[3.] there exists \(\mu^*\) such, that for \(\mu\leq\mu^*\) and \(\mu >\mu^*\) spreading or vanishing happens respectively.NEWLINENEWLINE\end{itemize}}