×

Sharp estimate of the spreading speed determined by nonlinear free boundary problems. (English) Zbl 1296.35219

A one-phase Stefan problem with unknown function \(u(t,x)\) and two free boundaries \(x = g(t)\) and \(x = h(t)\) is considred: \[ \begin{aligned} u_t - u_{xx} = f(u),\;t > 0, \;g(t) < x < h(t),\\ u(t,g(t)) = u(t,h(t)) = 0, \;g'(t) = -\mu u_x(t,g(t)), \;h'(t) = -\mu u_x(t,h(t)), \;t > 0, \\ g(0) = -h_0, \;h(0) = h_0, \;u(0,x) = u_0(x), \;-h_0 \leq x \leq h_0, \end{aligned}\tag{1} \] where \(\mu =\) const \(>0\), \(h_0 > 0\), the function \(f\) belongs to \(C^1([0,\infty))\), \(f(0) = 0\) and it is of monostable, bistable or combustion type.
The main result is as follows. Let the problem (1) have a unique classical solution \((u,\,g,\,h\)) and \(u \to 1, \;(g,\,h) \to (-\infty, \,\infty)\) as \(t \to \infty\). Suppose that for any \(\mu>0\) there exists a unique \(c^*>0\) such that the problem \(q'' - c^*q' +f(q)=0 \;\text{in} \;(0,\infty), \;q(0)=0,\;q(\infty)=1, \;q(z)>0 \;\text{in} \;(0,\infty)\) has a unique solution \(q_{c^*}(z)\), such that \(q'_{c^*}(0) = c^*/\mu\).
Then there exist \(\hat{H},\, \hat{G} \in \mathbb{R}\), such that for the solution of problem (1) the following asymptotic formulas hold: \[ \begin{aligned}\lim_{t\to\infty}\big(h(t)-c^*t- \hat{H}\big) =0, \;\lim_{t\to\infty}h'(t) = c^*, \;& \lim_{t\to\infty}\big(g(t)+ c^*t- \hat{G}\big) =0, \;\lim_{t\to\infty}g'(t) = - c^*, \\ \lim_{t\to\infty}\sup_{x\in[0,\, h(t)]} \big|u(t,x)-q_{c^*}(h(t) - x)\big| =0, & \;\lim_{t\to\infty}\sup_{x\in[g(t),\, 0]} \big|u(t,x)-q_{c^*}(x-g(t))\big| =0.\end{aligned} \]

MSC:

35R35 Free boundary problems for PDEs
35K58 Semilinear parabolic equations
35C07 Traveling wave solutions
35B40 Asymptotic behavior of solutions to PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
PDF BibTeX XML Cite
Full Text: DOI Link