Pulsating fronts for nonlocal dispersion and KPP nonlinearity

Coville, Jérôme; Dávila, Juan; Martínez, Salomé

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 2, pp. 179-223.

In this paper we are interested in propagation phenomena for nonlocal reaction–diffusion equations of the type:

$$\frac{\partial u}{\partial t}=J⁎u-u+f(x,u)t\in ℝ,x\in {ℝ}^N,$$

where J is a probability density and f is a KPP nonlinearity periodic in the x variables. Under suitable assumptions we establish the existence of pulsating fronts describing the invasion of the 0 state by a heterogeneous state. We also give a variational characterization of the minimal speed of such pulsating fronts and exponential bounds on the asymptotic behavior of the solution.

Classification : 45C05, 45G10, 45M15, 45M20, 92D25
Mots clés : Periodic front, Nonlocal dispersal, KPP nonlinearity

[1] G. Alberti, G. Bellettini, A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann. 310 no. 3 (1998), 527-560 | MR | Zbl

[2] P.W. Bates, P.C. Fife, X. Ren, X. Wang, Traveling waves in a convolution model for phase transition, Arch. Ration. Mech. Anal. 138 no. 2 (1997), 105-136 | MR | Zbl

[3] P.W. Bates, G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl. 332 no. 1 (2007), 428-440 | MR | Zbl

[4] H. Berestycki, F. Hamel, Gradient estimates for elliptic regularizations of semilinear parabolic and degenerate elliptic equations, Comm. Partial Differential Equations 30 no. 1–3 (2005), 139-156 | MR | Zbl

[5] H. Berestycki, F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math. 55 no. 8 (2002), 949-1032 | MR | Zbl

[6] H. Berestycki, F. Hamel, Generalized travelling waves for reaction–diffusion equations, Perspectives in Nonlinear Partial Differential Equations, Contemp. Math. vol. 446, Amer. Math. Soc., Providence, RI (2007), 101-123 | MR | Zbl

[7] H. Berestycki, F. Hamel, G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal. 255 no. 9 (2008), 2146-2189 | MR | Zbl

[8] H. Berestycki, F. Hamel, L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol. 51 no. 1 (2005), 75-113 | MR | Zbl

[9] H. Berestycki, F. Hamel, L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts, J. Math. Pures Appl. (9) 84 no. 8 (2005), 1101-1146 | MR | Zbl

[10] H. Berestycki, B. Larrouturou, P.-L. Lions, Multi-dimensional travelling-wave solutions of a flame propagation model, Arch. Ration. Mech. Anal. 111 no. 1 (1990), 33-49 | MR | Zbl

[11] H. Berestycki, L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 no. 5 (1992), 497-572 | EuDML | Numdam | MR | Zbl

[12] H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994), 47-92 | Zbl

[13] A. Brandt, Interior estimates for second-order elliptic differential (or finite-difference) equations via the maximum principle, Israel J. Math. 7 (1969), 95-121 | MR | Zbl

[14] A. Brandt, Interior Schauder estimates for parabolic differential- (or difference-) equations via the maximum principle, Israel J. Math. 7 (1969), 254-262 | MR | Zbl

[15] F.E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann. 142 (1960/1961), 22-130 | EuDML | MR | Zbl

[16] M.L. Cain, B.G. Milligan, A.E. Strand, Long-distance seed dispersal in plant populations, Am. J. Bot. 87 no. 9 (2000), 1217-1227

[17] J. Carr, A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132 no. 8 (2004), 2433-2439 | MR | Zbl

[18] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations 2 no. 1 (1997), 125-160 | MR | Zbl

[19] J.S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord, Am. Nat. 152 no. 2 (1998), 204-224

[20] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl. (4) 185 no. 3 (2006), 461-485 | MR | Zbl

[21] J. Coville, Travelling fronts in asymmetric nonlocal reaction diffusion equation: The bistable and ignition case, preprint of the CMM.

[22] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations 249 (2010), 2921-2953 | MR | Zbl

[23] J. Coville, L. Dupaigne, On a non-local reaction diffusion equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A 137 no. 4 (2007), 727-755 | MR | Zbl

[24] J. Coville, J. Dávila, S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal. 39 no. 5 (2008), 1693-1709 | MR | Zbl

[25] J. Coville, J. Dávila, S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations 244 no. 12 (2008), 3080-3118 | MR | Zbl

[26] A. De Masi, T. Gobron, E. Presutti, Travelling fronts in non-local evolution equations, Arch. Ration. Mech. Anal. 132 no. 2 (1995), 143-205 | MR | Zbl

[27] A. De Masi, E. Orlandi, E. Presutti, L. Triolo, Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics, Nonlinearity 7 (1994), 633-696 | MR | Zbl

[28] A. De Masi, E. Orlandi, E. Presutti, L. Triolo, Uniqueness and global stability of the instanton in nonlocal evolution equations, Rend. Mat. Appl. (7) 14 no. 4 (1994), 693-723 | MR | Zbl

[29] C. Deveaux, E. Klein, Estimation de la dispersion de pollen à longue distance à lʼechelle dʼun paysage agicole : une approche expérimentale, Publication du Laboratoire Ecologie, Systèmatique et Evolution, 2004.

[30] D.E. Edmunds, A.J.B. Potter, C.A. Stuart, Non-compact positive operators, Proc. R. Soc. Lond. Ser. A 328 no. 1572 (1972), 67-81 | MR | Zbl

[31] G.B. Ermentrout, J.B. Mcleod, Existence and uniqueness of travelling waves for a neural network, Proc. Roy. Soc. Edinburgh Sect. A 123 no. 3 (1993), 461-478 | MR | Zbl

[32] P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lect. Notes Biomath. vol. 28, Springer-Verlag, Berlin (1979) | MR | Zbl

[33] P.C. Fife, An integrodifferential analog of semilinear parabolic PDEs, Partial Differential Equations and Applications, Lect. Notes Pure Appl. Math. vol. 177, Dekker, New York (1996), 137-145 | MR | Zbl

[34] M.I. Freidlin, On wavefront propagation in periodic media, Stochastic Analysis and Applications, Adv. Probab. Relat. Top. vol. 7, Dekker, New York (1984), 147-166 | MR

[35] J. Gärtner, M.I. Freidlin, The propagation of concentration waves in periodic and random media, Dokl. Akad. Nauk SSSR 249 no. 3 (1979), 521-525 | MR

[36] J. García-Melián, J.D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal. 8 no. 6 (2009), 2037-2053 | MR | Zbl

[37] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer-Verlag, Berlin (2001) | MR | Zbl

[38] F. Hamel, L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. Eur. Math. Soc. (JEMS) 13 (2011), 345-390 | EuDML | MR | Zbl

[39] S. Heinze, Wave solutions to reaction–diffusion systems in perforated domains, Z. Anal. Anwend. 20 no. 3 (2001), 661-676 | MR | Zbl

[40] S. Heinze, G. Papanicolaou, A. Stevens, Variational principles for propagation speeds in inhomogeneous media, SIAM J. Appl. Math. 62 no. 1 (2001), 129-148 | MR | Zbl

[41] W. Hudson, B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media, Boundary Value Problems for Functional-Differential Equations, World Sci. Publ., River Edge, NJ (1995), 187-199 | Zbl

[42] V. Hutson, S. Martinez, K. Mischaikow, G.T. Vickers, The evolution of dispersal, J. Math. Biol. 47 no. 6 (2003), 483-517 | MR | Zbl

[43] B.F. Knerr, Parabolic interior Schauder estimates by the maximum principle, Arch. Ration. Mech. Anal. 75 no. 1 (1980/1981), 51-58 | MR | Zbl

[44] A.N. Kolmogorov, I.G. Petrovsky, N.S. Piskunov, Étude de lʼéquation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin Université dʼÉtat à Moscow, Série Internationale, Section A (1937), 1-26

[45] M. Kot, J. Medlock, Spreading disease: integro-differential equations old and new, Math. Biosci. 184 no. 2 (2003), 201-222 | MR | Zbl

[46] M.G. Krein, M.A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk (N.S.) 3 no. 1(23) (1948), 3-95 | MR | Zbl

[47] H. Matano, K.I. Nakamura, B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media 1 no. 4 (2006), 537-568 | MR | Zbl

[48] A. Mellet, J.-M. Roquejoffre, Y. Sire, Generalized fronts for one-dimensional reaction–diffusion equations, Discrete Contin. Dyn. Syst. A 26 no. 1 (2010), 303-312 | MR | Zbl

[49] J.D. Murray, Mathematical Biology, Biomathematics vol. 19, Springer-Verlag, Berlin (1993) | MR | Zbl

[50] G. Nadin, Traveling fronts in space–time periodic media, J. Math. Pures Appl. (9) 92 no. 3 (2009), 232-262 | MR | Zbl

[51] G. Nadin, L. Rossi, Propagation phenomena for time heterogeneous KPP reaction–diffusion equations, preprint. | MR

[52] J. Nolen, J.-M. Roquejoffre, L. Ryzhik, A. Zlatos, Existence and nonexistence of Fisher–KPP transition fronts, preprint.

[53] J. Nolen, L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 3 (2009), 1021-1047 | EuDML | Numdam | MR | Zbl

[54] R.D. Nussbaum, The radius of the essential spectrum, Duke Math. J. 37 (1970), 473-478 | MR | Zbl

[55] W.-X. Shen, Traveling waves in time dependent bistable equations, Differential Integral Equations 19 no. 3 (2006), 241-278 | MR | Zbl

[56] W.-X. Shen, A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations 249 no. 4 (2010), 747-795 | MR | Zbl

[57] W.-X. Shen, A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, http://arxiv.org/abs/1202.2452 | MR | Zbl

[58] N. Shigesada, K. Kawasaki, E. Teramoto, Traveling periodic waves in heterogeneous environments, Theor. Popul. Biol. 30 no. 1 (1986), 143-160 | MR | Zbl

[59] N. Shigesada, K. Kawasaki, E. Teramoto, The speeds of traveling frontal waves in heterogeneous environments, Lect. Notes Biomath. vol. 71, Springer, Berlin (1987), 88-97 | MR | Zbl

[60] F.M. Schurr, O. Steinitz, R. Nathan, Plant fecundity and seed dispersal in spatially heterogeneous environments: models, mechanisms and estimation, J. Ecol. 96 no. 4 (2008), 628-641

[61] H.F. Weinberger, On spreading speeds and travelling waves for growth and migration models in a periodic habitat, J. Math. Biol. 45 no. 6 (2002), 511-548 | MR | Zbl

[62] J. Xin, Existence of planar flame fronts in convective–diffusive periodic media, Arch. Ration. Mech. Anal. 121 no. 3 (1992), 205-233 | MR | Zbl

[63] J. Xin, Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity, J. Dynam. Differential Equations 3 no. 4 (1991), 541-573 | MR | Zbl

[64] J. Xin, Front propagation in heterogeneous media, SIAM Rev. 42 no. 2 (2000), 161-230 | MR | Zbl

[65] E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems, Springer-Verlag, New York (1986) | MR

[66] A. Zlatos, Generalized travelling waves in disordered media: Existence, uniqueness, and stability, preprint. | MR