# American Institute of Mathematical Sciences

May  2015, 35(5): 1843-1872. doi: 10.3934/dcds.2015.35.1843

## A variational approach to reaction-diffusion equations with forced speed in dimension 1

 1 Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France, France

Received  October 2013 Revised  October 2014 Published  December 2014

We investigate in this paper a scalar reaction diffusion equation with a nonlinear reaction term depending on $x-ct$. Here, $c$ is a prescribed parameter modelling the speed of climate change and we wonder whether a population will survive or not, that is, we want to determine the large-time behaviour of the associated solution.
This problem has been solved recently when the nonlinearity is of KPP type. We consider in the present paper general reaction terms, that are only assumed to be negative at infinity. Using a variational approach, we construct two thresholds $0<\underline{c}\leq \overline{c} <\infty$ determining the existence and the non-existence of travelling waves. Numerics support the conjecture $\underline{c}=\overline{c}$. We then prove that any solution of the initial-value problem converges at large times, either to $0$ or to a travelling wave. In the case of bistable nonlinearities, where the steady state $0$ is assumed to be stable, our results lead to constrasting phenomena with respect to the KPP framework. Lastly, we illustrate our results and discuss several open questions through numerics.
Citation: Juliette Bouhours, Grégroie Nadin. A variational approach to reaction-diffusion equations with forced speed in dimension 1. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1843-1872. doi: 10.3934/dcds.2015.35.1843
##### References:
 [1] H. Berestycki, L. Desvillettes and O. Diekmann, Can climate change lead to gap formation?,, in preparation., (). [2] H. Berestycki, O. Diekmann, C. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?,, Bull. Math. Biol., 71 (2009), 399. doi: 10.1007/s11538-008-9367-5. [3] H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains,, in preparation, (). doi: 10.1002/cpa.21536. [4] H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space,, Discrete Contin. Dyn. Syst., 21 (2008), 41. doi: 10.3934/dcds.2008.21.41. [5] H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed. II. Cylindrical-type domains,, Discrete Contin. Dyn. Syst., 25 (2009), 19. doi: 10.3934/dcds.2009.25.19. [6] K. J. Brown and H. Budin, On the existence of positive solutions for a class of semilinear elliptic boundary value problems,, SIAM J. Math. Anal., 10 (1979), 875. doi: 10.1137/0510082. [7] Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems,, J. Eur. Math. Soc. (JEMS), 12 (2010), 279. doi: 10.4171/JEMS/198. [8] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics,, 2nd edition, (2010). [9] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Ration. Mech. Anal., 65 (1977), 335. [10] R. Fisher, The advance of advantageous genes,, Annals of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. [11] T. Gallay and R. Joly, Global stability of travelling fronts for a damped wave equation with bistable nonlinearity,, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 103. [12] T. Gallay and E. Risler, A variational proof of global stability for bistable travelling waves,, Differential Integral Equations, 20 (2007), 901. [13] J. K. Hale and G. Raugel, Convergence in gradient-like systems with applications to PDE,, Z. Angew. Math. Phys., 43 (1992), 63. doi: 10.1007/BF00944741. [14] S. Heinze, A variational approach to traveling waves,, Technical Report 85, (). [15] A. Kolmogorov, I. Petrovskii and N. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de la matière et son application à un problème biologique,, Bull. Univ. Etat Mosc. Sér. Int. A, 1 (1937), 1. [16] M. Lucia, C. B. Muratov and M. Novaga, Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium,, Comm. Pure Appl. Math., 57 (2004), 616. doi: 10.1002/cpa.20014. [17] M. Lucia, C. B. Muratov and M. Novaga, Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders,, Arch. Ration. Mech. Anal., 188 (2008), 475. doi: 10.1007/s00205-007-0097-x. [18] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations,, J. Math. Kyoto Univ., 18 (1978), 221. [19] C. B. Muratov, A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 867. doi: 10.3934/dcdsb.2004.4.867. [20] A. B. Potapov and M. A. Lewis, Climate and competition: The effect of moving range boundaries on habitat invasibility,, Bull. Math. Biol., 66 (2004), 975. doi: 10.1016/j.bulm.2003.10.010. [21] E. Risler, Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure,, Ann. I. H. Poincaré, 25 (2008), 381. doi: 10.1016/j.anihpc.2006.12.005. [22] L. Roques, A. Roques, H. Berestycki and A. Kretzschmar, A population facing climate change: Joint influences of allee effects and environmental boundary geometry,, Population Ecology, 50 (2008), 215. doi: 10.1007/s10144-007-0073-1. [23] H. H. Vo, Traveling fronts for equations with forced speed in mixed environments,, in preparation., (). [24] T. Zelenyak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable,, Differentsial'nye Uravneniya, 4 (1968), 34. [25] Y. Zhou and M. Kot, Discrete-time growth-dispersal models with shifting species ranges,, Theor Ecol, 4 (2011), 13. doi: 10.1007/s12080-010-0071-3.

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##### References:
 [1] H. Berestycki, L. Desvillettes and O. Diekmann, Can climate change lead to gap formation?,, in preparation., (). [2] H. Berestycki, O. Diekmann, C. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?,, Bull. Math. Biol., 71 (2009), 399. doi: 10.1007/s11538-008-9367-5. [3] H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains,, in preparation, (). doi: 10.1002/cpa.21536. [4] H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space,, Discrete Contin. Dyn. Syst., 21 (2008), 41. doi: 10.3934/dcds.2008.21.41. [5] H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed. II. Cylindrical-type domains,, Discrete Contin. Dyn. Syst., 25 (2009), 19. doi: 10.3934/dcds.2009.25.19. [6] K. J. Brown and H. Budin, On the existence of positive solutions for a class of semilinear elliptic boundary value problems,, SIAM J. Math. Anal., 10 (1979), 875. doi: 10.1137/0510082. [7] Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems,, J. Eur. Math. Soc. (JEMS), 12 (2010), 279. doi: 10.4171/JEMS/198. [8] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics,, 2nd edition, (2010). [9] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Ration. Mech. Anal., 65 (1977), 335. [10] R. Fisher, The advance of advantageous genes,, Annals of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. [11] T. Gallay and R. Joly, Global stability of travelling fronts for a damped wave equation with bistable nonlinearity,, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 103. [12] T. Gallay and E. Risler, A variational proof of global stability for bistable travelling waves,, Differential Integral Equations, 20 (2007), 901. [13] J. K. Hale and G. Raugel, Convergence in gradient-like systems with applications to PDE,, Z. Angew. Math. Phys., 43 (1992), 63. doi: 10.1007/BF00944741. [14] S. Heinze, A variational approach to traveling waves,, Technical Report 85, (). [15] A. Kolmogorov, I. Petrovskii and N. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de la matière et son application à un problème biologique,, Bull. Univ. Etat Mosc. Sér. Int. A, 1 (1937), 1. [16] M. Lucia, C. B. Muratov and M. Novaga, Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium,, Comm. Pure Appl. Math., 57 (2004), 616. doi: 10.1002/cpa.20014. [17] M. Lucia, C. B. Muratov and M. Novaga, Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders,, Arch. Ration. Mech. Anal., 188 (2008), 475. doi: 10.1007/s00205-007-0097-x. [18] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations,, J. Math. Kyoto Univ., 18 (1978), 221. [19] C. B. Muratov, A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 867. doi: 10.3934/dcdsb.2004.4.867. [20] A. B. Potapov and M. A. Lewis, Climate and competition: The effect of moving range boundaries on habitat invasibility,, Bull. Math. Biol., 66 (2004), 975. doi: 10.1016/j.bulm.2003.10.010. [21] E. Risler, Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure,, Ann. I. H. Poincaré, 25 (2008), 381. doi: 10.1016/j.anihpc.2006.12.005. [22] L. Roques, A. Roques, H. Berestycki and A. Kretzschmar, A population facing climate change: Joint influences of allee effects and environmental boundary geometry,, Population Ecology, 50 (2008), 215. doi: 10.1007/s10144-007-0073-1. [23] H. H. Vo, Traveling fronts for equations with forced speed in mixed environments,, in preparation., (). [24] T. Zelenyak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable,, Differentsial'nye Uravneniya, 4 (1968), 34. [25] Y. Zhou and M. Kot, Discrete-time growth-dispersal models with shifting species ranges,, Theor Ecol, 4 (2011), 13. doi: 10.1007/s12080-010-0071-3.
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