## 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:

  D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974) Springer, Berlin, 1975, pp. 5 – 49. Lecture Notes in Math., Vol. 446. · Zbl 0325.35050  H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C, 569, H. Berestycki and Y. Pomeau eds., Kluwer, Dordrecht, 2003. · Zbl 1073.35113  J. Busca, M. A. Jendoubi, and P. Poláčik, Convergence to equilibrium for semilinear parabolic problems in \Bbb R^{\Bbb N}, Comm. Partial Differential Equations 27 (2002), no. 9-10, 1793 – 1814. · Zbl 1021.35013  Peter Constantin, Alexander Kiselev, and Leonid Ryzhik, Quenching of flames by fluid advection, Comm. Pure Appl. Math. 54 (2001), no. 11, 1320 – 1342. · Zbl 1032.35087  A. Fannjiang, A. Kiselev, and L. Ryzhik, Quenching of reaction by cellular flow, preprint. · Zbl 1097.35077  Paul C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal. 65 (1977), no. 4, 335 – 361. · Zbl 0361.35035  R.A. Fisher, The advance of advantageous genes, Ann. of Eugenics 7 (1937), 355-369. · JFM 63.1111.04  Ja. I. Kanel$$^{\prime}$$, Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sb. (N.S.) 59 (101) (1962), no. suppl., 245 – 288 (Russian).  Ja. I. Kanel$$^{\prime}$$, Stabilization of the solutions of the equations of combustion theory with finite initial functions, Mat. Sb. (N.S.) 65 (107) (1964), 398 – 413 (Russian).  A. Kiselev and A. Zlatos, Quenching of combustion by shear flows, to appear in Duke Math. J.  A.N. Kolmogorov, I.G. Petrovskii, and N.S. Piskunov, Étude de l’équation de la chaleur de matière et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh. 1 (1937), 1-25.  J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. Inst. Radio Eng. 50 (1962), 2061-2070.  Jean-Michel Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 4, 499 – 552 (English, with English and French summaries). · Zbl 0884.35013  Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. · Zbl 0807.35002  Jack X. Xin, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media, J. Statist. Phys. 73 (1993), no. 5-6, 893 – 926. · Zbl 1102.35340  Jack Xin, Front propagation in heterogeneous media, SIAM Rev. 42 (2000), no. 2, 161 – 230. · Zbl 0951.35060  J.B. Zel’dovich and D.A. Frank-Kamenetskii, A theory of thermal propagation of flame, Acta Physiochimica USSR 9 (1938), 341-350.  A. Zlatos, Quenching and propagation of combustion without ignition temperature cutoff, Nonlinearity 18 (2005), 1463-1475. · Zbl 1116.35069
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