Sharp transition between extinction and propagation of reaction

ZlatoÅ¡, Andrej¹

¹ Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Journal of the American Mathematical Society, Tome 19 (2006) no. 1, pp. 251-263

Résumé

We consider the reaction-diffusion equation \[ T_t = T_{xx} + f(T) \] on ${\mathbb {R}}$ with $T_0(x) \equiv \chi _{[-L,L]} (x)$ and $f(0)=f(1)=0$. In 1964 Kanel$^{\prime }$ proved that if $f$ is an ignition non-linearity, then $T\to 0$ as $t\to \infty$ when $L$, and $T\to 1$ when $L>L_1$. We answer the open question of the relation of $L_0$ and $L_1$ by showing that $L_0=L_1$. We also determine the large time limit of $T$ in the critical case $L=L_0$, thus providing the phase portrait for the above PDE with respect to a 1-parameter family of initial data. Analogous results for combustion and bistable non-linearities are proved as well.

Détail

DOI : 10.1090/S0894-0347-05-00504-7

Bibliographie

[1] Aronson, D. G., Weinberger, H. F. Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation 1975 5 49

[2] Busca, J., Jendoubi, M. A., Polã¡ÄIk, P. Convergence to equilibrium for semilinear parabolic problems in ℝ^{ℕ} Comm. Partial Differential Equations 2002 1793 1814

[3] Constantin, Peter, Kiselev, Alexander, Ryzhik, Leonid Quenching of flames by fluid advection Comm. Pure Appl. Math. 2001 1320 1342

[4] Fife, Paul C., Mcleod, J. B. The approach of solutions of nonlinear diffusion equations to travelling front solutions Arch. Rational Mech. Anal. 1977 335 361

[5] Kanel′, Ja. I. Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory Mat. Sb. (N.S.) 1962 245 288

[6] Kanel′, Ja. I. Stabilization of the solutions of the equations of combustion theory with finite initial functions Mat. Sb. (N.S.) 1964 398 413

[7] Roquejoffre, Jean-Michel Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders Ann. Inst. H. Poincaré C Anal. Non Linéaire 1997 499 552

[8] Smoller, Joel Shock waves and reaction-diffusion equations 1994

[9] Xin, Jack X. Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media J. Statist. Phys. 1993 893 926

[10] Xin, Jack Front propagation in heterogeneous media SIAM Rev. 2000 161 230