Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems

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May 2017, 22(3): 923-946. doi: 10.3934/dcdsb.2017047

Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems

Julián López-Gómez ^1,, , Marcela Molina-Meyer ^2, and Paul H. Rabinowitz ^3,

1.	Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain
2.	Departamento de Matemáticas, Universidad Carlos Ⅲ de Madrid, Legan´es 28071, Spain
3.	Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA, United States

* Corresponding author: Julián López-Gómez

This paper is dedicated to R.S.Cantrell on the occasion of his 60th birthday, for his pioneering work on the effects of spatial heterogeneities on nonlinear differential equations. With our friendship and best wishes for the future

Received July 2015 Revised June 2016 Published January 2017

Fund Project: Partially supported by grants MTM2012-30669 and MTM2015-65899-P of the Spanish Ministry of Economy and Competitiveness of Spain and the IMI of Complutense University.

In [12], the structure of the set of possible solutions of a degenerate boundary value problem was studied. For solutions with one interior zero, there were two possibilities for the solution set. In this paper, numerical examples are given showing each of these possibilities can occur.

Keywords: Degenerate boundary value problems, components of one node solutions.

Mathematics Subject Classification: Primary: 35J15, 35A15; Secondary: 35B38, 35B3.

Citation: Julián López-Gómez, Marcela Molina-Meyer, Paul H. Rabinowitz. Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 923-946. doi: 10.3934/dcdsb.2017047

References:

[1]	J. C. Eilbeck, The pseudo-spectral method and path-following in reaction-diffusion bifurcation studies, SIAM J. of Sci. Stat. Comput., 7 (1986), 599-610. doi: 10.1137/0907040. Google Scholar
[2]	J. M. Fraile, P. Koch, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Diff. Eqns., 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071. Google Scholar
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[5]	H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems Tata Institute of Fundamental Research, Springer, Berlin, 1987. Google Scholar
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[9]	J. López-Gómez, Metasolutions of Parabolic Equations in Population Dynamics CRC Press, Boca Raton, 2015. Google Scholar
[10]	J. López-Gómez, M. Molina-Meyer and A. Tellini, Spiraling bifurcation diagrams in superlinear indefinite problems, Disc. Cont. Dyn. Systems A, 35 (2015), 1561-1588. doi: 10.3934/dcds.2015.35.1561. Google Scholar
[11]	J. López-Gómez and P. H. Rabinowitz, The effects of spatial heterogeneities on some multiplicity results, Disc. Cont. Dyn. Systems A, 36 (2016), 941-952. doi: 10.3934/dcds.2016.36.941. Google Scholar
[12]	J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate boundary value problems, Adv. Nonl. Studies, 15 (2015), 253-288. doi: 10.1515/ans-2015-0201. Google Scholar
[13]	J. López-Gómez and A. Tellini, Generating an arbitrarily large number of isolas in a superlinear indefinite problem, Nonlinear Analysis, 108 (2014), 223-248. doi: 10.1016/j.na.2014.06.003. Google Scholar
[14]	H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 29 (1982), 401-441. Google Scholar
[15]	M. Molina-Meyer and F. R. Prieto-Medina, Numerical computation of classical and large solutions for the one-dimensional logistic equation with spatial heterogeneities, preprint. Google Scholar
[16]	T. Ouyang, On positive solutions of semilinear equations on compact manifolds, Ind. Math. J., 40 (1991), 1083-1141. doi: 10.1512/iumj.1991.40.40049. Google Scholar
[17]	P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Comm. Pure Appl. Math., 23 (1970), 939-961. doi: 10.1002/cpa.3160230606. Google Scholar
[18]	P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9. Google Scholar
[19]	P. H. Rabinowitz, A note on a nonlinear eigenvalue problem for a class of differential equations, J. Diff. Eqns., 9 (1971), 536-548. doi: 10.1016/0022-0396(71)90022-2. Google Scholar

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References:

[1]	J. C. Eilbeck, The pseudo-spectral method and path-following in reaction-diffusion bifurcation studies, SIAM J. of Sci. Stat. Comput., 7 (1986), 599-610. doi: 10.1137/0907040. Google Scholar
[2]	J. M. Fraile, P. Koch, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Diff. Eqns., 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071. Google Scholar
[3]	J. E. Furter and J. López-Gómez, Diffusion-mediated permanence problem for a heterogeneous Lotka-Volterra competition model, Proc. Royal Soc. Edinburgh, 127 (1997), 281-336. doi: 10.1017/S0308210500023659. Google Scholar
[4]	J. García-Melián, Multiplicity of positive solutions to boundary blow-up elliptic problems with sign changing weights, J. Funct. Anal., 261 (2011), 1775-1798. doi: 10.1016/j.jfa.2011.05.018. Google Scholar
[5]	H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems Tata Institute of Fundamental Research, Springer, Berlin, 1987. Google Scholar
[6]	J. López-Gómez, Approaching metasolutions by classical solutions, Differential and Integral Equations, 14 (2001), 739-750. Google Scholar
[7]	J. López-Gómez, Estabilidad y Bifurcación Estática. Aplicaciones y Métodos Numéricos Cuadernos de Matemática y Mecánica, Serie Cursos y Seminarios No 4, Santa Fe, 1988. Google Scholar
[8]	J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra, in Handbook of Differential Equations "Stationary Partial Differential Equations", (eds. M. Chipot and P. Quittner), North Holland, 2 (2005), 211–309. Google Scholar
[9]	J. López-Gómez, Metasolutions of Parabolic Equations in Population Dynamics CRC Press, Boca Raton, 2015. Google Scholar
[10]	J. López-Gómez, M. Molina-Meyer and A. Tellini, Spiraling bifurcation diagrams in superlinear indefinite problems, Disc. Cont. Dyn. Systems A, 35 (2015), 1561-1588. doi: 10.3934/dcds.2015.35.1561. Google Scholar
[11]	J. López-Gómez and P. H. Rabinowitz, The effects of spatial heterogeneities on some multiplicity results, Disc. Cont. Dyn. Systems A, 36 (2016), 941-952. doi: 10.3934/dcds.2016.36.941. Google Scholar
[12]	J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate boundary value problems, Adv. Nonl. Studies, 15 (2015), 253-288. doi: 10.1515/ans-2015-0201. Google Scholar
[13]	J. López-Gómez and A. Tellini, Generating an arbitrarily large number of isolas in a superlinear indefinite problem, Nonlinear Analysis, 108 (2014), 223-248. doi: 10.1016/j.na.2014.06.003. Google Scholar
[14]	H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 29 (1982), 401-441. Google Scholar
[15]	M. Molina-Meyer and F. R. Prieto-Medina, Numerical computation of classical and large solutions for the one-dimensional logistic equation with spatial heterogeneities, preprint. Google Scholar
[16]	T. Ouyang, On positive solutions of semilinear equations on compact manifolds, Ind. Math. J., 40 (1991), 1083-1141. doi: 10.1512/iumj.1991.40.40049. Google Scholar
[17]	P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Comm. Pure Appl. Math., 23 (1970), 939-961. doi: 10.1002/cpa.3160230606. Google Scholar
[18]	P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9. Google Scholar
[19]	P. H. Rabinowitz, A note on a nonlinear eigenvalue problem for a class of differential equations, J. Diff. Eqns., 9 (1971), 536-548. doi: 10.1016/0022-0396(71)90022-2. Google Scholar

Figure 1. The weight function $a=a_{0}$

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Figure 2. The weight function $a=a_\varepsilon $ for $\varepsilon >0$

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Figure 3. The metasolution $\boldsymbol{\mathfrak{m}}_{[(\frac{\pi}{h})^2, 1, 0]}$ for $a=a_{0}$

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Figure 4. A solution $u_{[{\rm{\lambda }}, 1, 0]}\sim \boldsymbol{\mathfrak{m}}_{[(\frac{\pi}{h})^2, 1, 0]}$

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Figure 5. The global bifurcation diagram for $\varepsilon=0.1$ and $0\leq {\rm{\lambda }} \leq 60$

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Figure 6. A series of solution plots on the principal curve for $\pi^2 < {\rm{\lambda }} < 400$ (left) and $450 < {\rm{\lambda }} < 700$ (right)

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Figure 7. A series of solutions on the isola for $20 < {\rm{\lambda }} < 40$

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Figure 8. A series of solutions on the isola for $70 < {\rm{\lambda }} < 140$

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Figure 9. The zeroes of the solutions computed for ${\rm{\lambda }}\leq 180$

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Figure 10. A zoom of the bifurcation diagram for $\varepsilon=0.001$

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Figure 11. Two significant components of the bifurcation diagram

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Figure 12. Two magnifications of the bifurcation diagram

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Figure 13. The zeroes of the solutions computed for $\varepsilon=0.0037$

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Figure 14. Two significant magnifications of the zeroes plots

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Figure 15. A series of solution plots along $\mathfrak{C}_2^+$

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Figure 16. A series of solution plots along ${\mathfrak{J}}^+$

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Figure 17. Solution plots along $\mathfrak{C}_2^+$

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Figure 18. Crossing the turning point of ${\mathfrak{J}}^+$

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Figure 19. Two components of the bifurcation diagram for $\varepsilon=0.0036$

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Figure 19">Figure 20. The two components plotted in Figure 19

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Figure 21. The zeroes of the solutions computed for $\varepsilon=0.0036$

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Figure 22. Two significant magnifications of the zeroes plots

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