Bistable reaction equations with doubly nonlinear diffusion

Alessandro Audrito

doi:10.3934/dcds.2019124

Article Contents

2019, Volume 39, Issue 6: 2977-3015. Doi: 10.3934/dcds.2019124

This issue Previous Article Next Article Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs

Bistable reaction equations with doubly nonlinear diffusion

Alessandro Audrito

Dipartimento di Matematica "Giuseppe Luigi Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Torino, Italia

Dedicated to Professor Juan Luis Vázquez

Received: January 2018

Revised: November 2018

Published: February 2019

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Abstract

Reaction-diffusion equations appear in biology and chemistry, and combine linear diffusion with different kind of reaction terms. Some of them are remarkable from the mathematical point of view, since they admit families of travelling waves that describe the asymptotic behaviour of a larger class of solutions $ 0\leq u(x, t)\leq 1 $ of the problem posed in the real line. We investigate here the existence of waves with constant propagation speed, when the linear diffusion is replaced by the "slow" doubly nonlinear diffusion. In the present setting we consider bistable reaction terms, which present interesting differences w.r.t. the Fisher-KPP framework recently studied in [5]. We find different families of travelling waves that are employed to describe the wave propagation of more general solutions and to study the stability/instability of the steady states, even when we extend the study to several space dimensions. A similar study is performed in the critical case that we call "pseudo-linear", i.e., when the operator is still nonlinear but has homogeneity one. With respect to the classical model and the "pseudo-linear" case, the travelling waves of the "slow" diffusion setting exhibit free boundaries.

Finally, as a complement of [5], we study the asymptotic behaviour of more general solutions in the presence of a "heterozygote superior" reaction function and doubly nonlinear diffusion ("slow" and "pseudo-linear").

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Mathematics Subject Classification: Primary: 35K57, 35K65; Secondary: 35C07.

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Figure 1. The "slow diffusion" area and the "pseudo-linear" line in $ (m, p-1) $-plane. The yellow and orange area are called "fast diffusion" and "very fast diffusion" range, respectively, and they will not be studied in this paper

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Figure 2. Qualitative representation of the reactions of type C and type C', respectively.

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Figure 3. Examples of admissible TWs: Finite and Positive types

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Figure 4. Reactions of type C, range $ \gamma > 0 $, case $ c = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3, 0.7 $. The second case is excluded by the assumption $ \int_0^1u^{m-1}f(u)du > 0 $

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Figure 6. Reactions of type C, range $ \gamma > 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $. The first two pictures show the case $ 0< c < c_{\ast} $, while the others the cases $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively

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Figure 5. Reactions of type C, range $ \gamma > 0 $. Null isoclines in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $, in the cases $ 0< c < c_0 $ and $ c > c_0 $, respectively

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Figure 7. Reactions of type C, range $ \gamma = 0$ , case $ c = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $.

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Figure 8. Reactions of type C, range $ \gamma = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $. The first picture shows the case $ 0< c < c_{\ast} $, while the others the cases $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively

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Figure 9. Reactions of type C', range $ \gamma > 0 $. Null isoclines in the $ (X, Z) $-plane for $ f(u) = u(1-u)(a - u) $, $ a = 0.3 $, in the cases $ 0< c < c_0 $ and $ c > c_0 $, respectively

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Figure 10. Reactions of type C', range $ \gamma > 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(a - u) $, $ a = 0.3 $, in the ranges $ 0< c < c_{\ast} $, $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively

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Figure 11. Reactions of type C', range $ \gamma = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(a - u) $, $ a = 0.3 $, in the ranges $ 0< c < c_{\ast} $, $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively

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