We deal with a reaction–diffusion equation u t = u xx + f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ1(x + c 1 t) (c 1 < 0) and ψ2(x + c 2 t) (c 2 > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all \((x, t) \in \mathbb{R}^{2}\). We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ1(x + c 1 t) and ψ2(x + c 2 t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c 1, we show the existence of an entire solution which behaves as ψ1( − x + c 1 t) in \(x\in(-\infty, (c_1+c)t/2]\) and φ(x + ct) in \(x\in[(c_1+c)t/2,\infty)\) for t≈ − ∞.
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References
Aronson D.G., Weinberger H.F. (1975). Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein J.A. (ed), Partial Differential Equations and Related Topics Lecture Notes in Math Vol 446. Springer, Berlin, pp. 5–49
Aronson D.G., Weinberger H.F. (1978). Multidimensional nonlinear diffusion arising in population genetics. Adv. math. 30, 33–76
Bramson M. (1983). Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Am. Math. Soc. 44, (285).
Chen X. (1997). Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Diff. Eq. 2, 125–160
Chen X., Guo J.-S. (2005). Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Diff. Eq. 212, 62–84
Ei S. (2000). The motion of weakly interacting pulses in reaction-diffusion systems. J. Dynam. Diff. Eq. 14, 85–137
Fife P.C., McLeod J.B. (1977). The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65, 335–361
Freidlin M. (1985). Limit theorems for large deviations and reaction-diffusion equations. Ann. Probab. 13, 639–675
Fukao Y., Morita Y., Ninomiya H. (2004). Some entire solutions of the Allen-Cahn equation. Taiwan. J. Math. 8, 15–32
Guo J.-S., Morita Y. (2005). Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations Discrete Contin. Dyn. Syst. 12, 193–212
Hamel F., Nadirashvili N. (1999). Entire solutions of the KPP equation. Commun. Pure Appl. Math. 52, 1255–1276
Kametaka Y. (1976). On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type. Osaka J. Math. 13, 11–66
Kawahara T., Tanaka M. (1983). Interactions of traveling fronts: An exact solutions of a nonlinear diffusion equations. Phys. Lett. 97A: 311–314
Kolmogorov A., Petrovsky I., Piskunov N. (1937). Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bjul. Moskowskogo Gos. Univ. Serv. Int. Secur. A 1, 1–26
Uchiyama K. (1978). The behavior of solutions of some nonlinear diffusion equations for large time. J. Math. Kyoto Univ. 18, 453–508
Yagisita H. (2003). Backward global solutions characterizing annihilation dynamics of travelling fronts. Public Res. Inst. Math. Sci. 39, 117–164
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Morita, Y., Ninomiya, H. Entire Solutions with Merging Fronts to Reaction–Diffusion Equations. J Dyn Diff Equat 18, 841–861 (2006). https://doi.org/10.1007/s10884-006-9046-x
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DOI: https://doi.org/10.1007/s10884-006-9046-x