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February  2020, 40(2): 1075-1105. doi: 10.3934/dcds.2020071

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

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

Received  May 2019 Revised  August 2019 Published  November 2019

Fund Project: 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

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.
Citation: Yu-Hao Liang, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1075-1105. doi: 10.3934/dcds.2020071
References:
[1]

T. BoddingtonP. Gray and C. Robinson, Thermal explosion and the disappearance of criticality at small activation energies: Exact results for the slab, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 368 (1979), 441-461.  doi: 10.1098/rspa.1979.0140.  Google Scholar

[2]

K. J. BrownM. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475-486.  doi: 10.1016/0362-546X(81)90096-1.  Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar

[4]

Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 173 (2001), 213-230.  doi: 10.1006/jdeq.2000.3932.  Google Scholar

[5]

J. Goddard II, Q. Morris, R. Shivaji and B. Son, Bifurcation curves for singular and nonsingular problems with nonlinear boundary conditions, Electron. J. Differential Equations, 2018 (2018), Paper No. 26, 12 pp.  Google Scholar

[6]

J. Goddard II, R. Shivaji and E. K. Lee, A double S-shaped bifurcation curve for a reaction-diffusion model with nonlinear boundary conditions, Bound. Value Probl., 2010 (2010), Art. ID 357542, 23 pp. doi: 10.1155/2010/357542.  Google Scholar

[7]

P. V. GordonE. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal.: Real World Appl., 15 (2014), 51-57.  doi: 10.1016/j.nonrwa.2013.05.005.  Google Scholar

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S.-Y. Huang and S.-H. Wang, On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion, Discret. Contin. Dyn. Syst., 35 (2015), 4839-4858.  doi: 10.3934/dcds.2015.35.4839.  Google Scholar

[9]

S.-Y. Huang and S.-H. Wang, Proof of a conjecture for the one-dimensional perturbed Gelfand problem from combustion theory, Arch. Ration. Mech. Anal., 222 (2016), 769-825.  doi: 10.1007/s00205-016-1011-1.  Google Scholar

[10]

K.-C. Hung and S.-H. Wang, A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differential Equations, 251 (2011), 223-237.  doi: 10.1016/j.jde.2011.03.017.  Google Scholar

[11]

K.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., 365 (2013), 1933-1956.  doi: 10.1090/S0002-9947-2012-05670-4.  Google Scholar

[12]

K.-C. HungS.-H. Wang and C.-H. Yu, Existence of a double $S$-shaped bifurcation curve with six positive solutions for a combustion problem, J. Math. Anal. Appl., 392 (2012), 40-54.  doi: 10.1016/j.jmaa.2012.02.036.  Google Scholar

[13]

P. Korman and Y. Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), 1011-1020.  doi: 10.1090/S0002-9939-99-04928-X.  Google Scholar

[14]

P. KormanY. Li and T. Ouyang, A simplified proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 263 (2017), 2874-2885.  doi: 10.1016/j.jde.2017.04.016.  Google Scholar

[15]

T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1–13. doi: 10.1512/iumj.1971.20.20001.  Google Scholar

[16]

Y.-H. Liang, S.-H. Wang, Detailed proofs of some results in the article: Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity, Available from: http://www.math.nthu.edu.tw/\symbol126shwang/SupplMatCubic.pdf. Google Scholar

[17]

Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions, J. Differential Equations, 260 (2016), 8358-8387.  doi: 10.1016/j.jde.2016.02.021.  Google Scholar

[18]

Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions Ⅱ, Electron. J. Differential Equations, 2017 (2017), Paper No. 61, 12 pp.  Google Scholar

[19]

M. Mimura and K. Sakamoto, Multi-dimensional transition layers for an exothermic reaction-diffusion system in long cylindrical domains, J. Math. Sci. Univ. Tokyo, 3 (1996), 109-179.   Google Scholar

[20]

J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.  doi: 10.1006/jfan.1999.3483.  Google Scholar

[21]

J. Shi, Multi-parameter bifurcation and applications, Topological Methods, Variational Methods and Their Applications (Taiyuan, 2002), World Sci. Publ., River Edge, NJ, 2003, 211–221.  Google Scholar

[22]

R. Shivaji, Remarks on an S-shaped bifurcation curve, J. Math. Anal. Appl., 111 (1985), 374-387.  doi: 10.1016/0022-247X(85)90223-9.  Google Scholar

[23]

C.-C. Tsai, S.-H. Wang and S.-Y. Huang, Classification and evolution of bifurcation curves for a one-dimensional Neumann-Robin problem and its applications, Electron. J. Qual. Theory Differ. Equ., 2018 (2018), Paper No. 85, 30 pp.  Google Scholar

[24]

C.-C. TzengK.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity, J. Differential Equations, 252 (2012), 6250-6274.  doi: 10.1016/j.jde.2012.02.020.  Google Scholar

[25]

S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), 1475-1485.  doi: 10.1016/0362-546X(94)90183-X.  Google Scholar

[26]

X. Zhang and M. Feng, Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps, Commun. Pure Appl. Anal., 17 (2018), 2149-2171.  doi: 10.3934/cpaa.2018103.  Google Scholar

show all references

References:
[1]

T. BoddingtonP. Gray and C. Robinson, Thermal explosion and the disappearance of criticality at small activation energies: Exact results for the slab, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 368 (1979), 441-461.  doi: 10.1098/rspa.1979.0140.  Google Scholar

[2]

K. J. BrownM. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475-486.  doi: 10.1016/0362-546X(81)90096-1.  Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar

[4]

Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 173 (2001), 213-230.  doi: 10.1006/jdeq.2000.3932.  Google Scholar

[5]

J. Goddard II, Q. Morris, R. Shivaji and B. Son, Bifurcation curves for singular and nonsingular problems with nonlinear boundary conditions, Electron. J. Differential Equations, 2018 (2018), Paper No. 26, 12 pp.  Google Scholar

[6]

J. Goddard II, R. Shivaji and E. K. Lee, A double S-shaped bifurcation curve for a reaction-diffusion model with nonlinear boundary conditions, Bound. Value Probl., 2010 (2010), Art. ID 357542, 23 pp. doi: 10.1155/2010/357542.  Google Scholar

[7]

P. V. GordonE. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal.: Real World Appl., 15 (2014), 51-57.  doi: 10.1016/j.nonrwa.2013.05.005.  Google Scholar

[8]

S.-Y. Huang and S.-H. Wang, On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion, Discret. Contin. Dyn. Syst., 35 (2015), 4839-4858.  doi: 10.3934/dcds.2015.35.4839.  Google Scholar

[9]

S.-Y. Huang and S.-H. Wang, Proof of a conjecture for the one-dimensional perturbed Gelfand problem from combustion theory, Arch. Ration. Mech. Anal., 222 (2016), 769-825.  doi: 10.1007/s00205-016-1011-1.  Google Scholar

[10]

K.-C. Hung and S.-H. Wang, A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differential Equations, 251 (2011), 223-237.  doi: 10.1016/j.jde.2011.03.017.  Google Scholar

[11]

K.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., 365 (2013), 1933-1956.  doi: 10.1090/S0002-9947-2012-05670-4.  Google Scholar

[12]

K.-C. HungS.-H. Wang and C.-H. Yu, Existence of a double $S$-shaped bifurcation curve with six positive solutions for a combustion problem, J. Math. Anal. Appl., 392 (2012), 40-54.  doi: 10.1016/j.jmaa.2012.02.036.  Google Scholar

[13]

P. Korman and Y. Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), 1011-1020.  doi: 10.1090/S0002-9939-99-04928-X.  Google Scholar

[14]

P. KormanY. Li and T. Ouyang, A simplified proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 263 (2017), 2874-2885.  doi: 10.1016/j.jde.2017.04.016.  Google Scholar

[15]

T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1–13. doi: 10.1512/iumj.1971.20.20001.  Google Scholar

[16]

Y.-H. Liang, S.-H. Wang, Detailed proofs of some results in the article: Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity, Available from: http://www.math.nthu.edu.tw/\symbol126shwang/SupplMatCubic.pdf. Google Scholar

[17]

Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions, J. Differential Equations, 260 (2016), 8358-8387.  doi: 10.1016/j.jde.2016.02.021.  Google Scholar

[18]

Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions Ⅱ, Electron. J. Differential Equations, 2017 (2017), Paper No. 61, 12 pp.  Google Scholar

[19]

M. Mimura and K. Sakamoto, Multi-dimensional transition layers for an exothermic reaction-diffusion system in long cylindrical domains, J. Math. Sci. Univ. Tokyo, 3 (1996), 109-179.   Google Scholar

[20]

J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.  doi: 10.1006/jfan.1999.3483.  Google Scholar

[21]

J. Shi, Multi-parameter bifurcation and applications, Topological Methods, Variational Methods and Their Applications (Taiyuan, 2002), World Sci. Publ., River Edge, NJ, 2003, 211–221.  Google Scholar

[22]

R. Shivaji, Remarks on an S-shaped bifurcation curve, J. Math. Anal. Appl., 111 (1985), 374-387.  doi: 10.1016/0022-247X(85)90223-9.  Google Scholar

[23]

C.-C. Tsai, S.-H. Wang and S.-Y. Huang, Classification and evolution of bifurcation curves for a one-dimensional Neumann-Robin problem and its applications, Electron. J. Qual. Theory Differ. Equ., 2018 (2018), Paper No. 85, 30 pp.  Google Scholar

[24]

C.-C. TzengK.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity, J. Differential Equations, 252 (2012), 6250-6274.  doi: 10.1016/j.jde.2012.02.020.  Google Scholar

[25]

S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), 1475-1485.  doi: 10.1016/0362-546X(94)90183-X.  Google Scholar

[26]

X. Zhang and M. Feng, Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps, Commun. Pure Appl. Anal., 17 (2018), 2149-2171.  doi: 10.3934/cpaa.2018103.  Google Scholar

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