Figure 1.
The weight function $a=a_{0}$
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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
| 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 |
In [
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 and Continuous Dynamical Systems - B,
2017, 22
(3)
: 923-946.
doi: 10.3934/dcdsb.2017047
![]()
References:
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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. |
| [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. |
| [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. |
| [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. |
| [5] |
H. B. Keller,
Lectures on Numerical Methods in Bifurcation Problems Tata Institute of Fundamental Research, Springer, Berlin, 1987. |
| [6] |
J. López-Gómez,
Approaching metasolutions by classical solutions, Differential and Integral Equations, 14 (2001), 739-750.
|
| [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. |
| [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. |
| [9] |
J. López-Gómez,
Metasolutions of Parabolic Equations in Population Dynamics CRC Press, Boca Raton, 2015. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
show all references
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. |
| [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. |
| [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. |
| [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. |
| [5] |
H. B. Keller,
Lectures on Numerical Methods in Bifurcation Problems Tata Institute of Fundamental Research, Springer, Berlin, 1987. |
| [6] |
J. López-Gómez,
Approaching metasolutions by classical solutions, Differential and Integral Equations, 14 (2001), 739-750.
|
| [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. |
| [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. |
| [9] |
J. López-Gómez,
Metasolutions of Parabolic Equations in Population Dynamics CRC Press, Boca Raton, 2015. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
Figure 4.
A solution $u_{[{\rm{\lambda }}, 1, 0]}\sim \boldsymbol{\mathfrak{m}}_{[(\frac{\pi}{h})^2, 1, 0]}$
Figure 6.
A series of solution plots on the principal curve for $\pi^2 < {\rm{\lambda }} < 400$ (left) and $450 < {\rm{\lambda }} < 700$ (right)
Figure 7.
A series of solutions on the isola for $20 < {\rm{\lambda }} < 40$
Figure 8.
A series of solutions on the isola for $70 < {\rm{\lambda }} < 140$
Figure 9.
The zeroes of the solutions computed for ${\rm{\lambda }}\leq 180$
Figure 10.
A zoom of the bifurcation diagram for $\varepsilon=0.001$
Figure 11.
Two significant components of the bifurcation diagram
Figure 12.
Two magnifications of the bifurcation diagram
Figure 13.
The zeroes of the solutions computed for $\varepsilon=0.0037$
Figure 14.
Two significant magnifications of the zeroes plots
Figure 15.
A series of solution plots along $\mathfrak{C}_2^+$
Figure 16.
A series of solution plots along ${\mathfrak{J}}^+$
Figure 17.
Solution plots along $\mathfrak{C}_2^+$
Figure 18.
Crossing the turning point of ${\mathfrak{J}}^+$
Figure 19.
Two components of the bifurcation diagram for $\varepsilon=0.0036$
Figure 20.
The two components plotted in Figure 19
Figure 21.
The zeroes of the solutions computed for $\varepsilon=0.0036$
Figure 22.
Two significant magnifications of the zeroes plots
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