Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-25T20:14:53.839Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence of bistable waves for a nonlocal and nonmonotone reaction-diffusion equation

Part of: Parabolic equations and systems Functional-differential and differential-difference equations

Published online by Cambridge University Press:  23 January 2019

Sergei Trofimchuk  and
Vitaly Volpert
Sergei Trofimchuk
Affiliation:
Instituto de Matemática y Fisica, Universidad de Talca, Talca, Chile
Vitaly Volpert
Affiliation:
Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France ([email protected]) and INRIA, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 Novembre 1918, 69200 Villeurbanne Cedex, France and People's Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation and Marchuk Institute of Numerical Mathematics of the RAS, ul. Gubkina 8, 119333 Moscow, Russian Federation
Get access Rights & Permissions [Opens in a new window]

Abstract

Reaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

Keywords

Reaction-diffusion equationnonlocalbistable nonlinearityexistence of wavesnonlinear Fredholm operatorLeray-Schauder method

MSC classification

Primary: 34K60: Qualitative investigation and simulation of models 35K57: Reaction-diffusion equations
Secondary: 92D25: Population dynamics (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 721 - 739
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alfaro, M., Ducrot, A. and Giletti, T.. Travelling waves for a non-monotone bistable equation with delay: existence and oscillations. Proc. London Math. Soc. 116 (2018), 729–759.CrossRefGoogle Scholar
2Apreutesei, N. and Volpert, V.. Properness and topological degree for nonlocal reaction-diffusion operators. Abstr. Appl. Anal. (2011), Art. ID 629692, 21 pp.Google Scholar
3Apreutesei, N. and Volpert, V.. Existence of travelling waves for a class of integro-differential equations from population dynamics. Int. Electron. J. Pure Appl. Math. 5 (2012), 53–67.Google Scholar
4Apreutesei, N., Ducrot, A. and Volpert, V.. Travelling waves for integro-differential equations in population dynamics. DCDS B 11 (2009), 541–561.CrossRefGoogle Scholar
5Bocharov, G., Meyerhans, A., Bessonov, N., Trofimchuk, S. and Volpert, V.. Spatiotemporal dynamics of virus infection spreading in tissues. PlosOne. 11 (2016), e0168576. https://doi.org/10.1371/journal.pone.0168576CrossRefGoogle ScholarPubMed
6Chen, X.. Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal evolution equations. Adv. Diff. Equ. 2 (1997), 125–160.Google Scholar
7Demin, I. and Volpert, V.. Existence of waves for a nonlocal reaction-diffusion equation. Math. Model. Nat. Phenom. 5 (2010), 80–101.CrossRefGoogle Scholar
8Ducrot, A., Marion, M. and Volpert, V.. Spectrum of some integro-differential operators and stability of travelling waves. Nonlin. Anal. TMA 74 (2011), 4455–4473.CrossRefGoogle Scholar
9Gomez, C., Prado, H. and Trofimchuk, S.. Separation dichotomy and wavefronts for a nonlinear convolution equation. J. Math. Anal. Appl. 420 (2014), 1–19.CrossRefGoogle Scholar
10Hasik, K., Kopfová, J., Nábělková, P. and Trofimchuk, S.. Traveling waves in the nonlocal KPP-Fisher equation: different roles of the right and the left interactions. J. Diff. Equ. 261 (2016), 1203–1236.Google Scholar
11Mallet-Paret, J.. The Fredholm alternative for functional differential equations of mixed type. J. Dynam. Diff. Equ. 11 (1999), 1–48.CrossRefGoogle Scholar
12Nadin, G., Rossi, L., Ryzhik, L. and Perthame, B.. Wave-like solutions for nonlocal reaction-diffusion equations: a toy model. Math. Model. Nat. Phenom. 8 (2013), 33–41.CrossRefGoogle Scholar
13Trofimchuk, S. and Volpert, V.. Global continuation of monotone waves for a unimodal bistable reaction-diffusion equation with delay, (2017) arXiv:1706.03403v1.Google Scholar
14Trofimchuk, S. and Volpert, V.. Travelling waves for a bistable reaction-diffusion equation with delay. SIAM J. Math. Anal. 50 (2018), 1175–1199.CrossRefGoogle Scholar
15Trofimchuk, E., Alvarado, P. and Trofimchuk, S.. On the geometry of wave solutions of a delayed reaction-diffusion equation. J. Diff. Equ. 246 (2009), 1422–1444.CrossRefGoogle Scholar
16Trofimchuk, E., Pinto, M. and Trofimchuk, S.. Monotone waves for non-monotone and non-local monostable reaction-diffusion equations. J. Diff. Equ. 261 (2016), 203–1236.CrossRefGoogle Scholar
17Volpert, V.. Elliptic partial differential equations, vol. 1. Fredholm theory of elliptic problems in unbounded domains (Basel: Birkhäuser, 2011).CrossRefGoogle Scholar
18Volpert, A. and Volpert, V.. Properness and topological degree for general elliptic operators. Abs. Appl. Anal. 2003 (2003), 129–181.CrossRefGoogle Scholar
19Volpert, A. I. and Volpert, V. A.. Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations. Trans. Moscow Math. Soc. 52 (1990), 59–108.Google Scholar
20Volpert, A., Volpert, V. and Volpert, V.L.. Traveling wave solutions of parabolic systems. Translation of Mathematical Monographs, vol. 140, (Providence: Amer. Math. Soc., 1994).CrossRefGoogle Scholar
21Wang, Z.-C., Li, W.-T. and Ruan, S.. Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay. J. Diff. Equ. 238 (2007), 153–200.CrossRefGoogle Scholar
22Wu, J. and Zou, X.. Traveling wave fronts of reaction-diffusion systems with delay. J. Dynam. Diff. Equ. 13 (2001), 651–687.CrossRefGoogle Scholar