# American Institute of Mathematical Sciences

June  2020, 40(6): 3031-3055. doi: 10.3934/dcds.2020053

## 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

* Corresponding author: Kousuke Kuto

Dedicated to Professor Wei-Ming Ni on the occasion of his 70th birthday

Received  April 2019 Revised  May 2019 Published  October 2019

Fund Project: The second author was supported in part by the Ministry of Science and Technology of Taiwan, grant 105-2115-M-007-009-MY3. The third author was partially supported by JSPS KAKENHI Grant-in-Aid Grant Numbers 15K04948 and 19K03581. The fourth author would like to thank the Mathematics Division of NCTS (Taipei Office) for the warm hospitality and the support of the fourth author's visit to Taiwan. The fourth author was partially supported by JSPS KAKENHI Grant Numbers JP26287024, JP15K04963, JP16K13778 and JP16KT0022

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 [11] and Stuart [13], to find solutions of elliptic partial differential equations with large amplitude. For the applications to the reaction-diffusion systems, we are able to relax the conditions to obtain the bifurcation from infinity for the following nonlinear terms; (ⅰ) nonlinear terms satisfying conditions similar to [11] (all directions), (ⅱ) nonlinear terms satisfying similar conditions only on the strip domain along the direction determined by the eigenfunction, (ⅲ) $p$-homogeneous nonlinear terms with degenerate conditions.

Citation: Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] J. Morgan, On a question of blow-up for semilinear parabolic systems, Differential Integral Equations, 3 (1990), 973-978.   Google Scholar [7] J. D. Murray, Mathematical Biology II : Spatial Models and Biomedical Applications, Third edition, Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] P. H. Rabinowitz, On bifurcation from infinity, J. Differential Equations, 14 (1973), 462-475.  doi: 10.1016/0022-0396(73)90061-2.  Google Scholar [12] S. Rosenblat and S. H. Davis, Bifurcation from infinity, SIAM Journal on Applied Mathematics, 37 (1979), 1-19.  doi: 10.1137/0137001.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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Dedicated to Professor Wei-Ming Ni on the occasion of his 70th birthday

##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] J. Morgan, On a question of blow-up for semilinear parabolic systems, Differential Integral Equations, 3 (1990), 973-978.   Google Scholar [7] J. D. Murray, Mathematical Biology II : Spatial Models and Biomedical Applications, Third edition, Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] P. H. Rabinowitz, On bifurcation from infinity, J. Differential Equations, 14 (1973), 462-475.  doi: 10.1016/0022-0396(73)90061-2.  Google Scholar [12] S. Rosenblat and S. H. Davis, Bifurcation from infinity, SIAM Journal on Applied Mathematics, 37 (1979), 1-19.  doi: 10.1137/0137001.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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