The Principal Eigenvalue of Cooperative Systems With Applications to a Model of Nonlinear Boundary Conditions

ABSTRACT In this paper, we study the eigenvalue problem for cooperative systems where the eigenvalue parameter appears on both the equation and the boundary. By utilizing a series of one‐parameter eigenvalue problems, we give a sufficient condition for the existence of the positive eigenvalue, which...

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Bibliographic Details
Published in:Studies in applied mathematics (Cambridge) Vol. 155; no. 5
Main Authors: Suriguga, Wu, Jianhua, Zhang, Lei
Format: Journal Article
Language:English
Published: Cambridge Blackwell Publishing Ltd 01.11.2025
Subjects:
bifurcation
Boundary conditions
cooperative systems
eigenvalue problem
Eigenvalues
Eigenvectors
nonlinear boundary condition
Parameters
ISSN:0022-2526, 1467-9590
Online Access:Get full text
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Summary:ABSTRACT In this paper, we study the eigenvalue problem for cooperative systems where the eigenvalue parameter appears on both the equation and the boundary. By utilizing a series of one‐parameter eigenvalue problems, we give a sufficient condition for the existence of the positive eigenvalue, which corresponds to the positive eigenfunction, and prove that it is unique when the system is symmetric. Then, we apply the theoretical result to investigate the existence and stability of non‐constant solutions for a general reaction‐diffusion model with nonlinear boundary conditions. In addition, the influence of nonlinear boundary conditions on the long‐time behavior of the solution is illustrated by numerical simulations.
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ISSN:0022-2526
1467-9590
DOI:10.1111/sapm.70148