Existence of travelling waves is studied for a delay reaction–diffusion system of equations describing the distribution of viruses and immune cells in the tissue. The proof uses the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Alfaro, M., Coville, J., Raoul, G.: Bistable travelling waves for nonlocal reaction diffusion equations. Discret. Contin. Dyn. Syst. 34, 1775–1791 (2014)
Alfaro, M., Ducrot, A.: Propagating interface in a monostable reaction-diffusion equation with time delay. Differ. Integral Equ. 27(1–2), 81–104 (2014)
Alfaro, M., Ducrot, A., Giletti, T.: Travelling waves for a non-monotone bistable equation with delay: existence and oscillations. Proc. London Math. Soc. 116(3), 729–759 (2018)
Bocharov, G., Meyerhans, A., Bessonov, N., Trofimchuk, S., Volpert, V.: Spatiotemporal dynamics of virus infection spreading in tissues. PLoS ONE 11(12), e0168576 (2016). https://doi.org/10.1371/journal.pone.0168576
Bocharov, G., Meyerhans, A., Bessonov, N., Trofimchuk, S., Volpert, V.: Modelling the dynamics of virus infection and immune response in space and time. Int. J. Parallel Emergent Distrib. Syst. (2017). https://doi.org/10.1080/17445760.2017.1363203
Bonnefon, O., Garnier, J., Hamel, F., Roques, L.: Inside dynamics of delayed traveling waves. Math. Mod. Nat. Phen. 8, 42–59 (2013)
Ducrot, A., Nadin, G.: Asymptotic behaviour of travelling waves for the delayed Fisher-KPP equation. J. Differ. Equ. 256(9), 3115–3140 (2014)
Fang, J., Zhao, X.-Q.: Bistable traveling waves for monotone semiflows with applications. J. Eur. Math. Soc. 17, 2243–2288 (2015)
Gomez, A., Trofimchuk, S.: Global continuation of monotone wavefronts. J. London Math. Soc. 89, 47–68 (2014)
Gourley, S.A., So, J.W.-H., Wu, J.H.: Non-locality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics. J. Math. Sci. 124, 5119–5153 (2004)
Ma, S., Wu, J.: Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation. J. Dynam. Differential Equations 19, 391–436 (2007)
Schaaf, K.: Asymptotic behavior and travelling wave solutions for parabolic functional differential equations. Trans. Amer. Math. Soc. 302, 587–615 (1987)
Smith, H.L., Zhao, X.-Q.: Global asymptotic stability of traveling waves in delayed reaction-diffusion equations. SIAM J. Math. Anal. 31, 514–534 (2000)
Trofimchuk, S., Volpert, V.: Travelling waves for a bistable reaction-diffusion equation with delay. SIAM J. Math. Anal. 50(1), 1175–1190 (2018)
Trofimchuk, S., Volpert, V.: Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities. Nonlinearity (2019). in press
Volpert, V.: Existence of viral infection waves in a model of immune response. In press
Volpert, A.I., 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, 59–108 (1990)
Volpert, A., Volpert, Vit., Volpert, Vl.: Traveling wave solutions of parabolic systems. Translation of mathematical monographs, Vol. 140, Amer. Math. Society, Providence (1994)
Volpert, A.I., Volpert, V.A.: The construction of the Leray-Schauder degree for elliptic operators in unbounded domains. Annales de l’IHP. Analyse non lineaire 11(3), 245–273 (1994)
Volpert, V., Volpert, A., Collet, J.F.: Topological degree for elliptic operators in unbounded cylinders. Adv. Differ. Equ. 4(6), 777–812 (1999)
Volpert, A., Volpert, V.: Properness and topological degree for general elliptic operators. Abstract Appl. Anal. 3, 129–181 (2003)
Wang, Z.-C., Li, W.-T., Ruan, S.: Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay. J. Differ. Equ. 238, 153–200 (2007)
The publication has been prepared with the support of the “RUDN University Program 5-100”, the Russian Science Foundation grant number 18-11-00171, and the French–Russian project PRC2307. The author is grateful to the anonymous reviewer for the profound reading of the paper and for the valuable remarks.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Volpert, V. Existence of Waves for a Bistable Reaction–Diffusion System with Delay. J Dyn Diff Equat 32, 615–629 (2020). https://doi.org/10.1007/s10884-019-09751-4
- Reaction–diffusion system
- Travelling wave
Mathematics subject classification