NONLINEAR STABILITY THROUGH ALGEBRAICALLY DECAYING POINT SPECTRUM: APPLICATIONS TO NONLOCAL INTERACTION EQUATIONS

作者：von Brecht James H^*; McCalla Scott G

来源：SIAM Journal on Mathematical Analysis, 2014, 46(6): 3727-3760.

DOI：10.1137/130949798

• nonlocal interactions
• nonlinear stability
• dynamical systems

Nonlocal interaction equations, such as aggregation equations and a number of related models for biological swarming, can exhibit compact, co-dimension one equilibrium solutions. Nonlinear stability of these solutions crucially depends on understanding the properties of operators whose linearizations have point spectrum that accumulates on the imaginary axis. In this paper, we establish criteria upon the linear operator, the nonlinearity, and the admissible perturbations under which linear stability is sufficient to guarantee nonlinear stability in the presence of such algebraically decaying point spectrum. We also provide temporal rates at which admissible perturbations decay to zero. We then apply the theory to some simplified nonlocal equations.

• 出版日期2014