American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 21-30. doi: 10.3934/proc.2013.2013.21

Global bifurcation diagrams of steady-states for a parabolic model related to a nuclear engineering problem

Inmaculada Antón 1, and  Julián López-Gómez 2,

1. 

Departamento de Matemática Aplicada, Escuela Universitaria de Estadística, Universidad Complutense de Madrid, 28040 Madrid, Spain
2. 

Departamento de Matematica Aplicada, Facultad de Ciencias Matematicas, Plaza de Ciencias 3, Universidad Complutense de Madrid, 28040-MADRID

Received  August 2012 Revised  May 2013 Published  November 2013

This paper studies the existence of coexistence states in a spatially heterogeneous reaction diffusion system arising in nuclear dynamics. Essentially, it establishes the existence of an unbounded component $\mathfrak{C}_+$ of the set of coexistence states of the system bifurcating from the trivial steady state solution, and it characterizes the values of the parameters where $\mathfrak{C}_+$ bifurcates from the trivial solution and from infinity. Throughout this paper, by a component it is meant a closed and connected subset which is maximal for the inclusion.
Keywords: bifurcation from infinity, nuclear dynamics, global bifurcation diagrams, non-cooperative systems., Coexistence states.
Mathematics Subject Classification: 35J57, 35J25, 35Q9.
Citation: Inmaculada Antón, Julián López-Gómez. Global bifurcation diagrams of steady-states for a parabolic model related to a nuclear engineering problem. Conference Publications, 2013, 2013 (special) : 21-30. doi: 10.3934/proc.2013.2013.21
References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions I,, Comm. Pure Appl. Maths. 12 (1959), 12 (1959), 623. Google Scholar

[2]

H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems,, Indiana Univ. Math. J. 21 (1971), 21 (1971), 125. Google Scholar

[3]

G. Arioli, Long term dynamics of a reaction-diffusion system,, J. Diff. Eqns. 235 (2007), 235 (2007), 298. Google Scholar

[4]

H. Brézis, Analyse Fontionnelle,, Masson, (1983). Google Scholar

[5]

S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems,, J. Diff. Eqns. 178 (2002), 178 (2002), 123. Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Anal. 8 (1971), 8 (1971), 321. Google Scholar

[7]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one,, Bull. London Math. Soc. 34 (2002), 34 (2002), 533. Google Scholar

[8]

J. Esquinas and J. López-Gómez, Optimal multiplicity in local birfurcation theory, I: Generalized generic eigenvalues,, J. Diff. Eqns. 71 (1988), 71 (1988), 72. Google Scholar

[9]

W. E. Kastenberg and P. L. Chambré, On the stability of nonlinear space-dependent reactor kinectics,, Nucl. Sci. Engrg. 31 (1968), 31 (1968), 67. Google Scholar

[10]

J. López-Gómez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems,, J. Diff. Eqns. 127 (1996), 127 (1996), 263. Google Scholar

[11]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis,, Chapman & Hall/CRC Research Notes in Mathematics 426, (2001). Google Scholar

[12]

J. López-Gómez, The steady-states of a non-cooperative model of nuclear reactors,, J. Diff. Eqns. 246 (2009), 246 (2009), 358. Google Scholar

[13]

J. López-Gómez, Elliptic Operators,, World Scientific Publishing, (2013). Google Scholar

[14]

J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications,, Diff. Int. Eqns. 7 (1994), 7 (1994), 383. Google Scholar

[15]

J. López-Gómez and M. Molina-Meyer, Bounded components of positive solutions of abstract fixed point equations: mushrooms, loops and isolas,, J. Diff. Eqns. 209 (2005), 209 (2005), 416. Google Scholar

[16]

J. López-Gómez and C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of Linear Operators,, Operator Theory: Advances and Applications Vol. 177, (2007). Google Scholar

[17]

P. de Mottoni and A. Tesei, Asymptotic stability for a system of quasilinear parabolic equations,, Appl. Ann. 9 (1979), 9 (1979), 7. Google Scholar

[18]

R. Peng, D. Wei and G. Yang, Asymptotic behaviour, uniqueness and stability of coexistence states of a non-cooperative reaction diffusion model of nuclear reactors,, Proc. Royal Soc. Edinburgh 140A (2010), 140A (2010), 189. Google Scholar

[19]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Funct. Anal. 7 (1971), 7 (1971), 487. Google Scholar

[20]

F. Rothe, Global solutions of R-D systems,, Springer, (1984). Google Scholar

[21]

W. Zhou, Uniqueness and asymptotic behavior of coexistence states for a non-cooperative model of nuclear reactors,, Nonl. Anal. TMA 72 (2010), 72 (2010), 2816. Google Scholar

show all references

References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions I,, Comm. Pure Appl. Maths. 12 (1959), 12 (1959), 623. Google Scholar

[2]

H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems,, Indiana Univ. Math. J. 21 (1971), 21 (1971), 125. Google Scholar

[3]

G. Arioli, Long term dynamics of a reaction-diffusion system,, J. Diff. Eqns. 235 (2007), 235 (2007), 298. Google Scholar

[4]

H. Brézis, Analyse Fontionnelle,, Masson, (1983). Google Scholar

[5]

S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems,, J. Diff. Eqns. 178 (2002), 178 (2002), 123. Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Anal. 8 (1971), 8 (1971), 321. Google Scholar

[7]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one,, Bull. London Math. Soc. 34 (2002), 34 (2002), 533. Google Scholar

[8]

J. Esquinas and J. López-Gómez, Optimal multiplicity in local birfurcation theory, I: Generalized generic eigenvalues,, J. Diff. Eqns. 71 (1988), 71 (1988), 72. Google Scholar

[9]

W. E. Kastenberg and P. L. Chambré, On the stability of nonlinear space-dependent reactor kinectics,, Nucl. Sci. Engrg. 31 (1968), 31 (1968), 67. Google Scholar

[10]

J. López-Gómez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems,, J. Diff. Eqns. 127 (1996), 127 (1996), 263. Google Scholar

[11]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis,, Chapman & Hall/CRC Research Notes in Mathematics 426, (2001). Google Scholar

[12]

J. López-Gómez, The steady-states of a non-cooperative model of nuclear reactors,, J. Diff. Eqns. 246 (2009), 246 (2009), 358. Google Scholar

[13]

J. López-Gómez, Elliptic Operators,, World Scientific Publishing, (2013). Google Scholar

[14]

J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications,, Diff. Int. Eqns. 7 (1994), 7 (1994), 383. Google Scholar

[15]

J. López-Gómez and M. Molina-Meyer, Bounded components of positive solutions of abstract fixed point equations: mushrooms, loops and isolas,, J. Diff. Eqns. 209 (2005), 209 (2005), 416. Google Scholar

[16]

J. López-Gómez and C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of Linear Operators,, Operator Theory: Advances and Applications Vol. 177, (2007). Google Scholar

[17]

P. de Mottoni and A. Tesei, Asymptotic stability for a system of quasilinear parabolic equations,, Appl. Ann. 9 (1979), 9 (1979), 7. Google Scholar

[18]

R. Peng, D. Wei and G. Yang, Asymptotic behaviour, uniqueness and stability of coexistence states of a non-cooperative reaction diffusion model of nuclear reactors,, Proc. Royal Soc. Edinburgh 140A (2010), 140A (2010), 189. Google Scholar

[19]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Funct. Anal. 7 (1971), 7 (1971), 487. Google Scholar

[20]

F. Rothe, Global solutions of R-D systems,, Springer, (1984). Google Scholar

[21]

W. Zhou, Uniqueness and asymptotic behavior of coexistence states for a non-cooperative model of nuclear reactors,, Nonl. Anal. TMA 72 (2010), 72 (2010), 2816. Google Scholar

[1]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242
[2]

Victor S. Kozyakin, Alexander M. Krasnosel’skii, Dmitrii I. Rachinskii. Arnold tongues for bifurcation from infinity. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 107-116. doi: 10.3934/dcdss.2008.1.107
[3]

Pablo Álvarez-Caudevilla, Julián López-Gómez. Characterizing the existence of coexistence states in a class of cooperative systems. Conference Publications, 2009, 2009 (Special) : 24-33. doi: 10.3934/proc.2009.2009.24
[4]

Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020053
[5]

Haiyan Wang, Carlos Castillo-Chavez. Spreading speeds and traveling waves for non-cooperative integro-difference systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2243-2266. doi: 10.3934/dcdsb.2012.17.2243
[6]

José Caicedo, Alfonso Castro, Arturo Sanjuán. Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1857-1865. doi: 10.3934/dcds.2017078
[7]

Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489
[8]

Michal Fečkan. Bifurcation from degenerate homoclinics in periodically forced systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 359-374. doi: 10.3934/dcds.1999.5.359
[9]

Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics. Conference Publications, 2013, 2013 (special) : 515-524. doi: 10.3934/proc.2013.2013.515
[10]

Tatsuki Mori, Kousuke Kuto, Masaharu Nagayama, Tohru Tsujikawa, Shoji Yotsutani. Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization. Conference Publications, 2015, 2015 (special) : 861-877. doi: 10.3934/proc.2015.0861
[11]

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
[12]

Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Spiraling bifurcation diagrams in superlinear indefinite problems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1561-1588. doi: 10.3934/dcds.2015.35.1561
[13]

E. Kapsza, Gy. Károlyi, S. Kovács, G. Domokos. Regular and random patterns in complex bifurcation diagrams. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 519-540. doi: 10.3934/dcdsb.2003.3.519
[14]

Po-Chun Huang, Shin-Hwa Wang, Tzung-Shin Yeh. Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2297-2318. doi: 10.3934/cpaa.2013.12.2297
[15]

Jean Dolbeault, Robert Stańczy. Bifurcation diagrams and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi--Dirac statistics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 139-154. doi: 10.3934/dcds.2015.35.139
[16]

Lora Billings, Erik M. Bollt, David Morgan, Ira B. Schwartz. Stochastic global bifurcation in perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 123-132. doi: 10.3934/proc.2003.2003.123
[17]

M. R. S. Kulenović, Orlando Merino. Global bifurcation for discrete competitive systems in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 133-149. doi: 10.3934/dcdsb.2009.12.133
[18]

S. Secchi, C. A. Stuart. Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493
[19]

Yilei Tang. Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2029-2046. doi: 10.3934/dcds.2018082
[20]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247

 Impact Factor: 

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences