Existence of traveling wave solutions to reaction-diffusion-ODE systems with hysteresis
This paper establishes the existence of traveling wave solutions to a reaction-diffusion equation coupled with a singularly perturbed first order ordinary differential equation with a small parameter ϵ>0. The system is a toy model for biological pattern formation. Traveling wave solutions corresp...
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| Main Authors: | , , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
15 August 2023
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| In: |
Journal of differential equations
Year: 2023, Volume: 364, Pages: 667-713 |
| ISSN: | 1090-2732 |
| DOI: | 10.1016/j.jde.2023.04.032 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1016/j.jde.2023.04.032 Verlag, lizenzpflichtig, Volltext: https://www.sciencedirect.com/science/article/pii/S0022039623003042 
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| Author Notes: | Lingling Hou, Hiroshi Kokubu, Anna Marciniak-Czochra, Izumi Takagi |
| Summary: | This paper establishes the existence of traveling wave solutions to a reaction-diffusion equation coupled with a singularly perturbed first order ordinary differential equation with a small parameter ϵ>0. The system is a toy model for biological pattern formation. Traveling wave solutions correspond to heteroclinic orbits of a fast-slow system. Under some conditions, the reduced problem (with ϵ=0) has a heteroclinic orbit with jump discontinuity, while the layer problem (i.e., the fast subsystem obtained as another limit of ϵ→0) has an orbit filling the gap. We thus construct a singular orbit by piecing together these two orbits. The traveling wave solution is obtained in the neighborhood of the singular orbit. However, unlike the classical FitzHugh-Nagumo equations, the singular orbit contains a fold point where the normal hyperbolicity breaks down and the standard Fenichel theory is not applicable. To circumvent this difficulty we employ the directional blowup method for geometric desingularization around the fold point. |
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| Item Description: | Online veröffentlicht am 9. Mai 2023 Gesehen am 30.11.2023 |
| Physical Description: | Online Resource |
| ISSN: | 1090-2732 |
| DOI: | 10.1016/j.jde.2023.04.032 |