Abstract
This article deals with the uniqueness and the behavior of solutions of non-local reaction diffusion equations. Since these equations share many properties with the usual reaction diffusion model, such as a form of maximum principle and the translation invariance, uniqueness and monotone behavior for the solution, as in the usual case, are expected. I present an elementary proof of this monotone behavior. The proof essentially uses techniques based on the maximum principle and the sliding method.
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Agmon, S., Nirenberg, L.: Properties of solutions of ordinary differential equations in Banach space. Commun. Pure Appl. Math. 16, 121–239 (1963)
Alikakos, N.D., Bates, P.W., Chen, X.: Periodic traveling waves and locating oscillating patterns in multidimensional domains. Trans. Am. Math. Soc. 351(7), 2777–2805 (1999)
Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30(1), 33–76 (1978)
Bates, P.W., Fife, P.C., Ren, X., Wang, X.: Travelling Waves in a convolution model for phase transition. Arch. Rat. Mech. Anal. 138, 105–136 (1997)
Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Commun. Pure Appl. Math. 55(8), 949–1032 (2002)
Berestycki, H., Larrouturou, B.: Quelques aspects mathématiques de la propagation des flammes prémélangées. (French) [Some mathematical aspects of premixed flame propagation] Nonlinear partial differential equations and their applications. Collége de France Seminar, Vol. X (Paris, 1987–1988), 65–129, Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow (1991)
Berestycki, H., Nirenberg, L.: Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations. J. Geom. Phys. 5(2), 237–275 (1988)
Berestycki, H., Nirenberg, L.: Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains. In: Rabinowitz, P.H., Zehnder, E. (eds.) Analysis, etc. (Volume Dedicated to J. Moser), pp. 497–572. Academic Press, Orlando, FL (1990)
Berestycki, H., Nirenberg, L.: Travelling fronts in cylinder. Ann. Inst. Henri Poincaré 9, 497–572 (1992)
Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bol. da Soc. Brasileira de Math. 22(1), 1–37 (1991)
Chen, X.: Existence, uniqueness and asymptotic stability of travelling fronts in non-local evolution equations. Adv. Differ. Equations 2, 125–160 (1997)
Coville, J.: équations de réaction-diffusion non-locale. Ph.D. Thesis, University of Paris 6, (2003)
Coville, J., Dupaigne, L.: Propagation speed of travelling fronts in non-local reaction diffusion equation. Nonlinear Anal. 60(5), 797–810 (2005)
De Masi, A., Gobron, E.P.: Travelling Fronts in non-local evolution equations. Arch. Rat. Mech. Anal. 132(2), 143–205 (1995)
Farina, A.: Monotonicity and one-dimensional symmetry for the solutions of Δ u+f(u)=0 in ℝN with possibly discontinuous nonlinearity. Adv. Math. Sci. Appl. 11(2), 811–834 (2001)
Fife, P.: Mathematical aspects of reacting and diffusing systems. Lecture Notes in Biomathematics, vol. 28. Springer, Berlin Heidelberg New York (1979)
Fife, P., McLeod, J.B.: The approach of solutions of nonlinear diffusion equation to travelling front solutions. Arch. Rat. Mech. Anal. 65, 335–361 (1977)
Fisher, R.A.: In: Bennett, J.H. ((ed.) with a foreword and notes) The Genetical Theory of Natural Selection—A Complete Variorum Edition (English summary). Revised reprint of the 1930 original edition. Oxford University Press, Oxford (1999)
Gilding, B.H., Kersner, R.: Travelling waves in nonlinear diffusion-convection reaction. Progress in Nonlinear Differential Equations and their Applications, vol. 60. Birkhäuser, Basel (2004)
Kolmogorov, A.N., Petrovsky, I.G., Piskunov, N.S.: étude de léquation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin Université détat à Moscow (Bjul. Moskowskogo Gos. Univ), Série Internationale, Section A. 1, 1937, pp. 1–26. English Translation: Study of the Difussion Equation with Growth of the Quantity of Matter and its Application to a Biological Problem. In Dynamics of curved front. R. Pelcé (ed.), Perspective in Physics Series pp. 105–130 Academic Press, New York (1988)
Murray, J.D.: Mathematical biology. Third edition. Interdisciplinary Applied Mathematics. vol. 17. Springer, Berlin Heidelberg New York (2002)
Vega, J.: On the uniqueness of multidimensional travelling fronts of some semilinear equations. J. Math. Anal. Appl. 177(2), 481–490 (1993)
Vega, J.: Travelling wavefronts of reaction-diffusion equations in cylindrical domains. Commun. Partial Differ. Equations 18(3–4), 505–531 (1993)
Zeldovich, J.B., Frank-Kamenetskii, D.A.: A Theory of Thermal Propagation of Flame. Acta Physiochimica U.R.S.S., Vol. 9 (1938). English Translation: In: Pelce, R. (ed.) Dynamics of Curved Fronts, Perspectives in Physics Series, pp. 131–140. Academic Press, New York (1988)
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Coville, J. On uniqueness and monotonicity of solutions of non-local reaction diffusion equation. Annali di Matematica 185, 461–485 (2006). https://doi.org/10.1007/s10231-005-0163-7
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DOI: https://doi.org/10.1007/s10231-005-0163-7