Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

Yu-Hao Liang; Shin-Hwa Wang

doi:10.3934/dcds.2020071

Article Contents

2020, Volume 40, Issue 2: 1075-1105. Doi: 10.3934/dcds.2020071

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Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

Yu-Hao Liang^1, , and
Shin-Hwa Wang^2,

1.
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
2.
Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan

^* Corresponding author: Yu-Hao Liang
^* Corresponding author: Yu-Hao Liang

Received: May 2019

Revised: August 2019

Published: November 2019

This work is partially supported by the Ministry of Science and Technology of the Republic of China under grant No. MOST 105-2115-M-007-002-MY3 and No. MOST 107-2115-M-390-006-MY2

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Abstract

We study the classification and evolution of bifurcation curves of positive solutions of the one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity given by

$ \left \{ \begin{array} [c]{l}u^{\prime \prime}(x)+\lambda(-\varepsilon u^{3}+u^{2}+u+1) = 0,\;0<x<1,\\ u(0) = 0,\ u^{\prime}(1) = -c<0, \end{array} \right. $

where $ 1/10\leq \varepsilon \leq1/5 $. It is interesting to find that the evolution of bifurcation curves is not completely identical with that for the one-dimensional perturbed Gelfand equations, even though it is the same for these two problems with zero Dirichlet boundary conditions. In fact, we prove that there exist a positive number $ \varepsilon^{\ast}\,(\approx0.178) $ and three nonnegative numbers $ c_{0}(\varepsilon)<c_{1}(\varepsilon)<c_{2}(\varepsilon) $ defined on $ [1/10,1/5] $ with $ c_{0} = 0 $ if $ 1/10<\varepsilon \leq \varepsilon^{\ast} $ and $ c_{0}>0 $ if $ \varepsilon^{\ast}<\varepsilon \leq1/5 $, such that, on the $ (\lambda,\Vert u\Vert_{\infty}) $-plane, (ⅰ) when $ 0<c\leq c_{0}(\varepsilon) $ and $ c\geq c_{2}(\varepsilon) $, the bifurcation curve is strictly increasing; (ⅱ) when $ c_{0}(\varepsilon)<c<c_{1}(\varepsilon) $, the bifurcation curve is $ S $-shaped; (ⅲ) when $ c_{1}(\varepsilon)\leq c<c_{2}(\varepsilon) $, the bifurcation curve is $ \subset $-shaped.

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Mathematics Subject Classification: 34B18, 74G35.

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Figure 1. Three different types of exactly $ S $-shaped bifurcation curves $ S_{c} $ with $ \lambda_{0}>0 $ and $ \left \Vert u_{\lambda_{0}}\right \Vert _{\infty}>0 $. (ⅰ) Type 1: $ \lambda_{0}<\lambda_{\ast}<\lambda^{\ast} $. (ⅱ) Type 2: $ \lambda_{0} = \lambda_{\ast}<\lambda^{\ast} $. (ⅲ) Type 3: $ \lambda_{\ast}<\lambda_{0}<\lambda^{\ast} $

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Figure 2. Exactly $ \subset $-shaped bifurcation curve $ S_{c} $ with $ \lambda _{0}>0 $ and $ \left \Vert u_{\lambda_{0}}\right \Vert _{\infty}>0 $

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Figure 3. Numerical simulations of bifurcation curves $ \bar{S} $ of (5) and $ \bar{S}_{c} $ of (6) with $ \bar{f}_{a}(u) = \exp \left( \frac{au}{a+u}\right) , $ $ a = 5 $ and with varying $ c>0 $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane of the bi-logarithm coordinates. Here $ c_{1,2}^{-}<c_{1,2}\,(\approx 0.488)<c_{1,2}^{+}<c_{1}\,(\approx1.365)<c_{1}^{+}<c_{2}\,(\approx 7.718)<c_{2}^{+}<c_{3}\,(\approx47.711)<c_{3}^{+} $ (adopted from [17,Fig. 4])

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Figure 4. Numerical simulations of bifurcation curves $ S $ of (2) and $ S_{c} $ of (1) with $ \varepsilon = 1/10 $ and varying $ c>0 $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane. Here $ c_{1,2}^{-}<c_{1,2}\,(\approx0.384)<c_{1,2}^{+} <c_{1}\,(\approx1.117)<c_{1}^{+}<c_{2}\,(\approx39.438)<c_{2}^{+} $ and $ \beta_{1/10}\approx10.992 $ (cf. Fig. 3)

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Figure 5. Numerical simulation of the bifurcation surface $\Gamma \equiv \left \{ (\lambda,c,\left \Vert u_{\lambda,c}\right \Vert _{\infty}):\lambda,c>0\text{ and }u_{\lambda,c}\text{ is a positive solution of (1) )}\right \} $ with $ \varepsilon = 1/10 $ in the $ (\lambda,c,\left \Vert u\right \Vert _{\infty}) $-space. (cf. Fig. 4)

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Figure 6. Numerical simulations of bifurcation curves $ S $ of (2) and $ S_{c} $ of (1) with $ \varepsilon = 1/5 $ and varying $ c>0 $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane. Here $ c_{0}^{-}<c_{0}\,(\approx0.121)<c_{1,2}^{-} <c_{1,2}\,(\approx0.609)<c_{1,2}^{+}<c_{1}\,(\approx1.120)<c_{1}^{+} <c_{2}\,(\approx19.052)<c_{2}^{+} $ and $ \beta_{1/5}\approx5.977 $

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Figure 7. Numerical simulation of the bifurcation surface $\Gamma \equiv \left \{ (\lambda,c,\left \Vert u_{\lambda,c}\right \Vert _{\infty}):\lambda,c>0\text{ and }u_{\lambda,c}\text{ is a positive solution of (1)}\right \} $ with $ \varepsilon = 1/5 $ in the $ (\lambda,c,\left \Vert u\right \Vert _{\infty}) $-space. (cf. Fig. 6)

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Figure 8. Numerical simulations of bifurcation curves $ S $ and $ S_{c} $ with $ \varepsilon = 12/5 $ and varying $ c>0 $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane. Here $ \beta_{12/5}\approx1.120 $

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Figure 9. Numerical simulations of bifurcation curves $ S $ of (2) and $ S_{c} $ of (1) with $ \varepsilon = 7/4 $ and varying $ c>0 $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane. Here $ c_{0}^{-}<c_{0}\,(\approx1.432)<c_{0}^{+} <c_{2}\,(\approx2.382)<c_{2}^{+} $ and $ \beta_{7/4}\approx1.327 $

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Figure 10. (a): The conjectured classification of bifurcation curves $ S_{c} = S_{\varepsilon,c} $ of (1) on the first quadrant of $ (\varepsilon,c) $-plane. (b)–(i): Distinct shapes of bifurcation curves $ S_{c} $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane in the different partitions of the first quadrant of $ (\varepsilon,c) $-plane

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Figure 11. Numerical simulations of bifurcation curves $ \hat{S} $ of (52) and $ \hat{S}_{c} $ of (51) with $ \varepsilon = 1/10 $ and varying $ c>0 $ on the $ (\lambda,\left \Vert u\right \Vert _{\infty}) $-plane. Here $ c_{0,1} ^{-}<c_{0,1}(\approx0.750)<c_{0,1}^{+}<c_{1}\,(\approx4.009)<c_{1}^{+} <c_{2}\,(\approx28.174)<c_{2,3}^{-}<c_{2,3}\,(\approx28.570)<c_{2,3}^{+} <c_{3}\,(\approx28.727)<c_{3}^{+}\, $ and $ \hat{\beta}_{1/10}\approx9.014 $

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