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

Du, Yihong; Matsuzawa, Hiroshi; Zhou, Maolin

SIAM J. Math. Anal. 46, No. 1, 375-396 (2014).

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: \[ \begin{aligned}\lim_{t\to\infty}\big(h(t)-c^*t- \hat{H}\big) =0, \;\lim_{t\to\infty}h'(t) = c^*, \;& \lim_{t\to\infty}\big(g(t)+ c^*t- \hat{G}\big) =0, \;\lim_{t\to\infty}g'(t) = - c^*, \\ \lim_{t\to\infty}\sup_{x\in[0,\, h(t)]} \big|u(t,x)-q_{c^*}(h(t) - x)\big| =0, & \;\lim_{t\to\infty}\sup_{x\in[g(t),\, 0]} \big|u(t,x)-q_{c^*}(x-g(t))\big| =0.\end{aligned} \]

Reviewer: Galina Bizhanova (Almaty)

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

Keywords:

nonlinear free boundary problems; Stefan problem; semilinear parabolic equation; traveling wave solution; asymptotic behavior of solutions