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Bistable reaction equations with doubly nonlinear diffusion

Dedicated to Professor Juan Luis Vázquez

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  • 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").

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

    Citation:

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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

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

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

    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 $

    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

    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

    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 $.

    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

    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

    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

    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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