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Preface: DCDS-A special issue to honor Wei-Ming Ni's 70th birthday
Bifurcation from infinity with applications to reaction-diffusion systems
1. | Nakano Junior and Senior High School Attached to Meiji University, 3-3-4 Higashi-Nakano, Nakano-ku, 164-0003, Japan |
2. | Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan |
3. | Department of Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan |
4. | School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan |
The bifurcation method is one of powerful tools to study the existence of a continuous branch of solutions. However without further analysis, the local theory only ensures the existence of solutions within a small neighborhood of bifurcation point. In this paper we extend the theory of bifurcation from infinity, initiated by Rabinowitz [
References:
[1] |
C.-N. Chen,
Uniqueness and bifurcation for solutions of nonlinear Sturm-Liouville eigenvalue problems, Arch. Rational. Mech. Anal., 111 (1990), 51-85.
doi: 10.1007/BF00375700. |
[2] |
C.-N. Chen,
Some existence and bifurcation results for solutions of nonlinear Sturm-Liouville eigenvalue problems, Math. Zeitschrift, 208 (1991), 177-192.
doi: 10.1007/BF02571519. |
[3] |
C.-N. Chen, A survey of nonlinear Sturm-Liouville equations, Sturm-Liouville Theory, Birkhäuser, Basel, (2005), 201–216.
doi: 10.1007/3-7643-7359-8_9. |
[4] |
M. Fila and K. Ninomiya,
Reaction versus diffusion: Blow-up induced and inhibited by diffusivity, Russian Mathematical Surveys, 60 (2005), 1217-1235.
doi: 10.1070/RM2005v060n06ABEH004289. |
[5] |
N. Mizoguchi, H. Ninomiya and E. Yanagida,
Diffusion-induced blowup in a nonlinear parabolic system, J. Dynam. Differential Equations, 10 (1998), 619-638.
doi: 10.1023/A:1022633226140. |
[6] |
J. Morgan,
On a question of blow-up for semilinear parabolic systems, Differential Integral Equations, 3 (1990), 973-978.
|
[7] |
J. D. Murray, Mathematical Biology II : Spatial Models and Biomedical Applications, Third edition, Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. |
[8] |
H. Ninomiya and H. F. Weinberger,
Pest control may make the pest population explode, Z. Angew. Math. Phys., 54 (2003), 869-873.
doi: 10.1007/s00033-003-3210-5. |
[9] |
H. Ninomiya and H. F. Weinberger,
On p-homogeneous systems of differential equations and their linear perturbations, Applicable Analysis, 85 (2006), 225-247.
doi: 10.1080/0036810500277066. |
[10] |
P. H. Rabinowitz,
Some global results for nonlinear eigenvalue problems, Journal of functional analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[11] |
P. H. Rabinowitz,
On bifurcation from infinity, J. Differential Equations, 14 (1973), 462-475.
doi: 10.1016/0022-0396(73)90061-2. |
[12] |
S. Rosenblat and S. H. Davis,
Bifurcation from infinity, SIAM Journal on Applied Mathematics, 37 (1979), 1-19.
doi: 10.1137/0137001. |
[13] |
C. A. Stuart,
Solutions of large norm for non-linear Sturm-Liouville problems, Quarterly Journal of Mathematics, 24 (1973), 129-139.
doi: 10.1093/qmath/24.1.129. |
[14] |
A. M. Turing,
The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. Ser. B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
show all references
References:
[1] |
C.-N. Chen,
Uniqueness and bifurcation for solutions of nonlinear Sturm-Liouville eigenvalue problems, Arch. Rational. Mech. Anal., 111 (1990), 51-85.
doi: 10.1007/BF00375700. |
[2] |
C.-N. Chen,
Some existence and bifurcation results for solutions of nonlinear Sturm-Liouville eigenvalue problems, Math. Zeitschrift, 208 (1991), 177-192.
doi: 10.1007/BF02571519. |
[3] |
C.-N. Chen, A survey of nonlinear Sturm-Liouville equations, Sturm-Liouville Theory, Birkhäuser, Basel, (2005), 201–216.
doi: 10.1007/3-7643-7359-8_9. |
[4] |
M. Fila and K. Ninomiya,
Reaction versus diffusion: Blow-up induced and inhibited by diffusivity, Russian Mathematical Surveys, 60 (2005), 1217-1235.
doi: 10.1070/RM2005v060n06ABEH004289. |
[5] |
N. Mizoguchi, H. Ninomiya and E. Yanagida,
Diffusion-induced blowup in a nonlinear parabolic system, J. Dynam. Differential Equations, 10 (1998), 619-638.
doi: 10.1023/A:1022633226140. |
[6] |
J. Morgan,
On a question of blow-up for semilinear parabolic systems, Differential Integral Equations, 3 (1990), 973-978.
|
[7] |
J. D. Murray, Mathematical Biology II : Spatial Models and Biomedical Applications, Third edition, Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. |
[8] |
H. Ninomiya and H. F. Weinberger,
Pest control may make the pest population explode, Z. Angew. Math. Phys., 54 (2003), 869-873.
doi: 10.1007/s00033-003-3210-5. |
[9] |
H. Ninomiya and H. F. Weinberger,
On p-homogeneous systems of differential equations and their linear perturbations, Applicable Analysis, 85 (2006), 225-247.
doi: 10.1080/0036810500277066. |
[10] |
P. H. Rabinowitz,
Some global results for nonlinear eigenvalue problems, Journal of functional analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[11] |
P. H. Rabinowitz,
On bifurcation from infinity, J. Differential Equations, 14 (1973), 462-475.
doi: 10.1016/0022-0396(73)90061-2. |
[12] |
S. Rosenblat and S. H. Davis,
Bifurcation from infinity, SIAM Journal on Applied Mathematics, 37 (1979), 1-19.
doi: 10.1137/0137001. |
[13] |
C. A. Stuart,
Solutions of large norm for non-linear Sturm-Liouville problems, Quarterly Journal of Mathematics, 24 (1973), 129-139.
doi: 10.1093/qmath/24.1.129. |
[14] |
A. M. Turing,
The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. Ser. B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
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