# American Institute of Mathematical Sciences

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

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.
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. [2] H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), 125-146. [3] G. Arioli, Long term dynamics of a reaction-diffusion system, J. Diff. Eqns. 235 (2007), 298-307. [4] H. Brézis, Analyse Fontionnelle, Masson, Paris, 1983. [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. [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340. [7] E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc. 34 (2002), 533-538. [8] J. Esquinas and J. López-Gómez, Optimal multiplicity in local birfurcation theory, I: Generalized generic eigenvalues, J. Diff. Eqns. 71 (1988), 72-92. [9] W. E. Kastenberg and P. L. Chambré, On the stability of nonlinear space-dependent reactor kinectics, Nucl. Sci. Engrg. 31 (1968), 67-79. [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. [11] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman & Hall/CRC Research Notes in Mathematics 426, Boca Raton, Florida, 2001. [12] J. López-Gómez, The steady-states of a non-cooperative model of nuclear reactors, J. Diff. Eqns. 246 (2009), 358-372. [13] J. López-Gómez, Elliptic Operators, World Scientific Publishing, Singapore, 2013. [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. [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. [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. [17] P. de Mottoni and A. Tesei, Asymptotic stability for a system of quasilinear parabolic equations, Appl. Ann. 9 (1979), 7-21. [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. [19] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513. [20] F. Rothe, Global solutions of R-D systems, Springer, 1984. [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.

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##### 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. [2] H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), 125-146. [3] G. Arioli, Long term dynamics of a reaction-diffusion system, J. Diff. Eqns. 235 (2007), 298-307. [4] H. Brézis, Analyse Fontionnelle, Masson, Paris, 1983. [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. [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340. [7] E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc. 34 (2002), 533-538. [8] J. Esquinas and J. López-Gómez, Optimal multiplicity in local birfurcation theory, I: Generalized generic eigenvalues, J. Diff. Eqns. 71 (1988), 72-92. [9] W. E. Kastenberg and P. L. Chambré, On the stability of nonlinear space-dependent reactor kinectics, Nucl. Sci. Engrg. 31 (1968), 67-79. [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. [11] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman & Hall/CRC Research Notes in Mathematics 426, Boca Raton, Florida, 2001. [12] J. López-Gómez, The steady-states of a non-cooperative model of nuclear reactors, J. Diff. Eqns. 246 (2009), 358-372. [13] J. López-Gómez, Elliptic Operators, World Scientific Publishing, Singapore, 2013. [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. [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. [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. [17] P. de Mottoni and A. Tesei, Asymptotic stability for a system of quasilinear parabolic equations, Appl. Ann. 9 (1979), 7-21. [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. [19] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513. [20] F. Rothe, Global solutions of R-D systems, Springer, 1984. [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.
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