## 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

### Keywords:

reaction-diffusion equations; one space dimension
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### References:

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