N-barrier maximum principle for degenerate elliptic systems and its application

Chiun-Chuan Chen; Li-Chang Hung; Hsiao-Feng Liu

doi:10.3934/dcds.2018034

Article Contents

2018, Volume 38, Issue 2: 791-821. Doi: 10.3934/dcds.2018034

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N-barrier maximum principle for degenerate elliptic systems and its application

1.
Department of Mathematics, National Taiwan University, National Center for Theoretical Sciences, Taipei, Taiwan
2.
College of Engineering, National Taiwan University of Science and Technology, Department of Mathematics, National Taiwan University, Taipei, Taiwan
3.
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan

* Corresponding author
* Corresponding author

Received: October 31, 2016

Revised: August 31, 2017

Published: November 2017

The research of C.-C. Chen is partly supported by the grant 102-2115-M-002-011-MY3 of Ministry of Science and Technology, Taiwan. The research of L.-C. Hung is partly supported by the grant 104EFA0101550 of Ministry of Science and Technology, Taiwan

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Abstract

In this paper, we prove the N-barrier maximum principle, which extends the result in C.-C. Chen and L.-C. Hung (2016) from linear diffusion equations to nonlinear diffusion equations, for a wide class of degenerate elliptic systems of porous medium type. The N-barrier maximum principle provides a priori upper and lower bounds of the solutions to the above-mentioned degenerate nonlinear diffusion equations including the Shigesada-Kawasaki-Teramoto model as a special case. We also apply the N-barrier maximum principle to a coexistence problem in ecology, where we show the nonexistence of traveling waves in a three-species degenerate elliptic system.

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Mathematics Subject Classification: Primary: 35B50; Secondary: 35C07, 35K57.

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Figure 1. Red line: $1-u-a_1\, v=0$; blue line: $1-a_2\, u-v=0$; green curve: $F(u, v):=\alpha\, u\, (1-u-a_1\, v)+\beta\, v\, (1-a_2\, u-v)=0$; brown line: $\displaystyle\frac{u}{\underline{u}}+\frac{v}{\underline{v}}=1$, where $\underline{u}$ and $\underline{v}$ are given by (55) and (56); magenta ellipse (above): $\alpha\, d_1\, u^2+\beta\, d_2\, v^2=\lambda_1$, where $\lambda_1$ is given by (61); magenta ellipse (below): $\alpha\, d_1\, u^2+\beta\, d_2\, v^2=\lambda_2$, where $\lambda_2$ is given by (70); yellow line (above): $\alpha\, u+\beta\, v=\eta_1$, where $\eta_1$ is given by (62); yellow line (below): $\alpha\, u+\beta\, v=\eta_2$, where $\eta_2$ is given by (71); dashed orange curve: the solution $(u(x), v(x))$; dotted line (above): $\displaystyle\frac{u}{\sqrt{\frac{\lambda_1}{\alpha\, d_1}}}+\displaystyle\frac{v}{\sqrt{\frac{\lambda_1}{\beta\, d_2}}}=1$; dotted line (below): $\displaystyle\frac{u}{\sqrt{\frac{\lambda_2}{\alpha\, d_1}}}+\displaystyle\frac{v}{\sqrt{\frac{\lambda_2}{\beta\, d_2}}}=1$

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Figure 2. Red line: $1-u-a_1\, v=0$; blue line: $1-a_2\, u-v=0$; green curve: $F(u, v)=0$; brown line: $\displaystyle\frac{u}{\overline{u}}+\frac{v}{\overline{v}}=1$, where $\overline{u}$ and $\overline{v}$ are given by (55) and (56); magenta ellipses : $\alpha\, d_1\, u^2+\beta\, d_2\, v^2=\lambda_1, \lambda_2$, where $\lambda_1$ (below) is given by (96) and $\lambda_2$ (above) by (103); yellow lines: $\alpha\, u+\beta\, v=\eta_1, \eta_2$, where $\eta_1$ (below) is given by (102) and $\eta_2$ (above) by (109); dashed orange curve: the solution $(u(x), v(x))$; dotted lines: $\displaystyle\sqrt{\alpha\, d_1}\, u+\sqrt{\beta\, d_2}\, v=\sqrt{\lambda_1}$ (below), $\displaystyle\sqrt{\lambda_2}$ (above); $\overline{u}=\overline{v}=1$; $d_1=3$, $a_1=2$, $a_2=3$, $\alpha=1$

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Figure 3. Red: $u(x) =60\, \big(1-\tanh x\big)^2$; green: $v(x) =8\, \big(1+\tanh x\big)$

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Figure 4. Red: $u(x) =\displaystyle\frac{1}{10}\big(1-\cos\, (2\, x)\big)$; green: $v(x) =\displaystyle\frac{1}{11}\big(1+\cos\, (2\, x)\big)$; blue: $w(x) =\displaystyle\frac{1}{12}\big(1+\cos\, (2\, x)\big)$

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