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

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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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  • 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.

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

    Citation:

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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): $\displaystyle\frac{u}{\sqrt{\frac{\lambda_2}{\alpha\, d_1}}}+\displaystyle\frac{v}{\sqrt{\frac{\lambda_2}{\beta\, d_2}}}=1$

    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); $\overline{u}=\overline{v}=1$; $d_1=3$, $a_1=2$, $a_2=3$, $\alpha=1$

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

    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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