Abstract
Existence of travelling waves is studied for a 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.
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Notes
The system is locally monotone if the derivatives in (1.7) are positive. This case can be obtained from the case with negative derivatives by a change of variables.
References
Bocharov, G.A.: Modelling the dynamics of LCMV infection in mice: conventional and exhaustive CTL responses. J. Theor. Biol. 192, 283–308 (1998)
Bocharov, G., Argilaguet, J., Meyerhans, A.: Understanding experimental LCMV infection of mice: the role of mathematical models. J. Immunol. Res. 2015, 739706 (2015)
Bocharov, G., Meyerhans, A., Bessonov, N., Trofimchuk, S., Volpert, V.: Spatiotemporal dynamics of virus infection spreading in tissues. Plos One (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 Emerg. Distrib. Syst. 1, 1 (2017). https://doi.org/10.1080/17445760.2017.1363203
Fitzpatrick, P.M., Pejsachowicz, J., Rabier, P.J.: The degree of proper \(C^2\) Fredholm mappings. J. Reine Angew. 427, 1–33 (1992)
Fitzpatrick, P.M., Pejsachowicz, J., Rabier, P.J.: Orientability of Fredholm families and topological degree for orientable nonlinear Fredholm mappings. J. Funct. Anal. 124, 1–39 (1994)
Trofimchuk, S., Volpert, V.: Travelling waves for a bistable reaction-diffusion equation with delay. arxiv:1701.08560v1
Volpert, A.I., Volpert, V.A.: Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations. Trans. Mosc. Math. Soc. 52, 59–108 (1990)
Volpert, A.: Vit. Volpert, Vl. Volpert. In: Traveling Wave Solutions of Parabolic Systems. Translation of Mathematical Monographs, vol. 140. American Mathematical 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)
Volpert, V.: Elliptic Partial Differential Equations, vol. 1. Fredholm Theory of Elliptic Problems in Unbounded Domains. Birkhäuser, Basel (2011)
Volpert, V.: Elliptic Partial Differential Equations: Volume 2: Reaction-Diffusion Equations. Birkhäuser, Basel (2014)
Acknowledgements
The publication has been prepared with the support of the “RUDN University Program 5-100” and the French–Russian project PRC2307.
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Volpert, V. Existence of Reaction–Diffusion Waves in a Model of Immune Response. Mediterr. J. Math. 17, 47 (2020). https://doi.org/10.1007/s00009-020-1490-z
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DOI: https://doi.org/10.1007/s00009-020-1490-z