Electron. J. Differential Equations, Vol. 2021 (2021), No. 76, pp. 1-21.

Traveling waves for unbalanced bistable equations with density dependent diffusion

Pavel Drabek, Michaela Zahradnikova

Abstract:
We study the existence and qualitative properties of traveling wave solutions for the unbalanced bistable reaction-diffusion equation with a rather general density dependent diffusion coefficient. In particular, it allows for singularities and/or degenerations as well as discontinuities of the first kind at a finite number of points. The reaction term vanishes at equilibria and it is a continuous, possibly non-Lipschitz function. We prove the existence of a unique speed of propagation and a unique traveling wave profile (up to translation) which is a non-smooth function in general. In the case of the power-type behavior of the diffusion and reaction near equilibria we provide detailed asymptotic analysis of the profile.

Submitted August 22, 2021. Published September 14, 2021.
Math Subject Classifications: 35Q92, 35C07, 34A12, 35K92.
Key Words: Density dependent diffusion; unbalanced bistable reaction term; degenerate and singular diffusion; traveling wave; degenerate non-Lipschitz reaction.

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Pavel Drábek
Department of Mathematics, Faculty of Applied Sciences
University of West Bohemia
Univerzitní 8, 30100 Plzen, Czech Republic
email: pdrabek@kma.zcu.cz
Michaela Zahradníková
Department of Mathematics, Faculty of Applied Sciences
University of West Bohemia
Univerzitní 8, 30100 Plzen, Czech Republic
email: mzahrad@kma.zcu.cz

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