American Institute of Mathematical Sciences

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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), 623-727.  Google Scholar

[2]

H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), 125-146.  Google Scholar

[3]

G. Arioli, Long term dynamics of a reaction-diffusion system, J. Diff. Eqns. 235 (2007), 298-307.  Google Scholar

[4]

H. Brézis, Analyse Fontionnelle, Masson, Paris, 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), 123-211.  Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340.  Google Scholar

[7]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc. 34 (2002), 533-538.  Google Scholar

[8]

J. Esquinas and J. López-Gómez, Optimal multiplicity in local birfurcation theory, I: Generalized generic eigenvalues, J. Diff. Eqns. 71 (1988), 72-92.  Google Scholar

[9]

W. E. Kastenberg and P. L. Chambré, On the stability of nonlinear space-dependent reactor kinectics, Nucl. Sci. Engrg. 31 (1968), 67-79. 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), 263-294.  Google Scholar

[11]

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

[12]

J. López-Gómez, The steady-states of a non-cooperative model of nuclear reactors, J. Diff. Eqns. 246 (2009), 358-372.  Google Scholar

[13]

J. López-Gómez, Elliptic Operators, World Scientific Publishing, Singapore, 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), 383-398.  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), 416-441.  Google Scholar

[16]

J. López-Gómez and C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of Linear Operators, Operator Theory: Advances and Applications Vol. 177, Birkhäuser, Springer, Basel-Boston-Berlin, 2007.  Google Scholar

[17]

P. de Mottoni and A. Tesei, Asymptotic stability for a system of quasilinear parabolic equations, Appl. Ann. 9 (1979), 7-21.  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), 189-201.  Google Scholar

[19]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513.  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), 2816-2820.  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), 623-727.  Google Scholar

[2]

H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), 125-146.  Google Scholar

[3]

G. Arioli, Long term dynamics of a reaction-diffusion system, J. Diff. Eqns. 235 (2007), 298-307.  Google Scholar

[4]

H. Brézis, Analyse Fontionnelle, Masson, Paris, 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), 123-211.  Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340.  Google Scholar

[7]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc. 34 (2002), 533-538.  Google Scholar

[8]

J. Esquinas and J. López-Gómez, Optimal multiplicity in local birfurcation theory, I: Generalized generic eigenvalues, J. Diff. Eqns. 71 (1988), 72-92.  Google Scholar

[9]

W. E. Kastenberg and P. L. Chambré, On the stability of nonlinear space-dependent reactor kinectics, Nucl. Sci. Engrg. 31 (1968), 67-79. 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), 263-294.  Google Scholar

[11]

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

[12]

J. López-Gómez, The steady-states of a non-cooperative model of nuclear reactors, J. Diff. Eqns. 246 (2009), 358-372.  Google Scholar

[13]

J. López-Gómez, Elliptic Operators, World Scientific Publishing, Singapore, 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), 383-398.  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), 416-441.  Google Scholar

[16]

J. López-Gómez and C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of Linear Operators, Operator Theory: Advances and Applications Vol. 177, Birkhäuser, Springer, Basel-Boston-Berlin, 2007.  Google Scholar

[17]

P. de Mottoni and A. Tesei, Asymptotic stability for a system of quasilinear parabolic equations, Appl. Ann. 9 (1979), 7-21.  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), 189-201.  Google Scholar

[19]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513.  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), 2816-2820.  Google Scholar

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