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Sharp transition between extinction and propagation of reaction. (English) Zbl 1081.35011
Concerning the following one-dimensional reaction-diffusion equation \[ u_t=u_{xx}+f(u),\quad x\in{\mathbb R}, t>0. (*) \] the author proves two long-time convergence results for certain nonlinearities \(f\) and initial values of the special type of characteristic functions on intervals. One of them reads as follows:
Theorem 1. Let \(\theta_0\in [0,1)\) and \(f:[0,1]\to{\mathbb R}\) be Lipschitz with \(f(1)=0\) and such that \(f(\theta)=0\) for all \(\theta\in [0,\theta_0],\) and \(f(\theta)>0\) for all \(\theta\in (\theta_0,1).\) If \(\theta_0>0,\) then assume in addition that \(f\) is nondecreasing on \([\theta_0,\theta_0+\delta]\) with some \(\delta>0.\) Let \(u(x,t)\) be a global solution of \((*)\) with the initial value \(u(x,0)=\chi_{[-L,L]}(x).\) Then there exists a constant \(L_0\geq 0\) with the following properties: (i) if \(L<L_0,\) then \(u(x,t)\to 0\) uniformly on \({\mathbb R}\) as \(t\to\infty.\) (ii) if \(L=L_0,\) then \(u(x,t)\to\theta_0\) uniformly on compact intervals as \(t\to\infty.\) (iii) if \(L>L_0,\) then \(u(x,t)\to 1\) uniformly on compact intervals as \(t\to\infty.\)

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
35K15 Initial value problems for second-order parabolic equations
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