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., (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics,, 2nd edition, (2010). Google Scholar

[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. Google Scholar

[10]

R. Fisher, The advance of advantageous genes,, Annals of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[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. Google Scholar

[12]

T. Gallay and E. Risler, A variational proof of global stability for bistable travelling waves,, Differential Integral Equations, 20 (2007), 901. Google Scholar

[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. Google Scholar

[14]

S. Heinze, A variational approach to traveling waves,, Technical Report 85, (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations,, J. Math. Kyoto Univ., 18 (1978), 221. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

H. H. Vo, Traveling fronts for equations with forced speed in mixed environments,, in preparation., (). Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

H. Berestycki, L. Desvillettes and O. Diekmann, Can climate change lead to gap formation?,, in preparation., (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics,, 2nd edition, (2010). Google Scholar

[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. Google Scholar

[10]

R. Fisher, The advance of advantageous genes,, Annals of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[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. Google Scholar

[12]

T. Gallay and E. Risler, A variational proof of global stability for bistable travelling waves,, Differential Integral Equations, 20 (2007), 901. Google Scholar

[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. Google Scholar

[14]

S. Heinze, A variational approach to traveling waves,, Technical Report 85, (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations,, J. Math. Kyoto Univ., 18 (1978), 221. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

H. H. Vo, Traveling fronts for equations with forced speed in mixed environments,, in preparation., (). Google Scholar

[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. Google Scholar

[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. Google Scholar

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