# Pulsating solutions for multidimensional bistable and multistable equations

- 4 Downloads

## Abstract

We investigate the existence of pulsating front-like solutions for spatially periodic heterogeneous reaction–diffusion equations in arbitrary dimension, in both bistable and more general multistable frameworks. In the multistable case, the notion of a single front is not sufficient to understand the dynamics of solutions, and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states whose speeds are ordered. Surprisingly, for a given equation, the shape of this terrace (i.e., the involved intermediate states or even the cardinality of the family of fronts) may depend on the direction of propagation.

## Notes

### Acknowledgements

This study was funded by European Research Council (No. 321186) and Agence Nationale de la Recherche (No. ANR-14-CE25-0013).

## References

- 1.Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math.
**30**(1), 33–76 (1978)MathSciNetCrossRefGoogle Scholar - 2.Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Commun. Pure Appl. Math.
**55**(8), 949–1032 (2002)MathSciNetCrossRefGoogle Scholar - 3.Berestycki, H., Hamel, F.: Generalized transition waves and their properties. Commun. Pure Appl. Math.
**65**(5), 592–648 (2012)MathSciNetCrossRefGoogle Scholar - 4.Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model. I. Species persistence. J. Math. Biol.
**51**(1), 75–113 (2005)MathSciNetCrossRefGoogle Scholar - 5.Dancer, E.N., Hess, P.: Stability of fixed points for order-preserving discrete-time dynamical systems. J. Reine Angew. Math.
**419**, 125–139 (1991)MathSciNetzbMATHGoogle Scholar - 6.Ducrot, A.: A multi-dimensional bistable nonlinear diffusion equation in a periodic medium. Math. Ann.
**366**, 783–818 (2016)MathSciNetCrossRefGoogle Scholar - 7.Ducrot, A., Giletti, T., Matano, H.: Existence and convergence to a propagating terrace in one-dimensional reaction–diffusion equations. Trans. Am. Math. Soc.
**366**(10), 5541–5566 (2014)MathSciNetCrossRefGoogle Scholar - 8.Ding, W., Hamel, F., Zhao, X.-Q.: Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat. Indiana Univ. Math. J.
**66**(4), 1189–1265 (2017)MathSciNetCrossRefGoogle Scholar - 9.Fang, J., Zhao, X.-Q.: Bistable traveling waves for monotone semiflows with applications. J. Eur. Math. Soc.
**17**(9), 2243–2288 (2015)MathSciNetCrossRefGoogle Scholar - 10.Fife, P.C., McLeod, J.: The approach of solutions of nonlinear diffusion equations to traveling front solutions. Arch. Ration. Mech. Anal.
**65**, 335–361 (1977)MathSciNetCrossRefGoogle Scholar - 11.Giletti, T., Matano, H.: Existence and uniqueness of propagating terraces. Commun. Contemp. Math.
**(to appear)**Google Scholar - 12.Hamel, F.: Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity. J. Math. Pures Appl.
**89**, 355–399 (2008)MathSciNetCrossRefGoogle Scholar - 13.Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer, Berlin (1976)Google Scholar
- 14.Li, B., Weinberger, H.F., Lewis, M.A.: Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci.
**196**(1), 82–98 (2005)MathSciNetCrossRefGoogle Scholar - 15.Matano, H.: Existence of nontrivial unstable sets for equilibriums of strongly ordered-preserving systems. J. Fac. Sci. Univ. Kyoto
**30**, 645–673 (1984)zbMATHGoogle Scholar - 16.Nadin, G.: The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator. SIAM J. Math. Anal.
**4**, 2388–2406 (2010)MathSciNetCrossRefGoogle Scholar - 17.Poláčik, P.: Propagating terraces and the dynamics of front-like solutions of reaction–diffusion equations on \(\mathbb{R}\). Mem. Am. Math. Soc.
**(to appear)**Google Scholar - 18.Risler, E.: Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure. Ann. Inst. H. Poincaré Anal. Non Linéaire
**25**(2), 381–424 (2008)MathSciNetCrossRefGoogle Scholar - 19.Weinberger, H.F.: On spreading speeds and traveling waves for growth and migration models in a periodic habitat. J. Math. Biol.
**45**(6), 511–548 (2002)MathSciNetCrossRefGoogle Scholar - 20.Xin, J.: Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity. J. Dyn. Differ. Equ.
**3**, 541–573 (1991)MathSciNetCrossRefGoogle Scholar