Adiabatic Stability Under Semi-Strong Interactions: The Weakly Damped Regime

作者：Bellsky Thomas^*; Doelman Arjen; Kaper Tasso J; Promislow Keith

来源：Indiana University Mathematics Journal, 2013, 62(6): 1809-1859.

DOI：10.1512/iumj.2013.62.5159

• Reaction-diffusion system
• semi-strong interaction
• renormalization group
• nonlocal eigenvalue problem (NLEP)
• normal hyperbolicity

We rigorously derive multi-pulse interaction laws for the semi-strong interactions in a family of singularly-perturbed and weakly-damped reaction-diffusion systems in one space dimension. Most significantly, we show the existence of a manifold of quasi-steady N-pulse solutions and identify a %26quot;normal-hyperbolicity%26quot; condition which balances the asymptotic weakness of the linear damping against the algebraic evolution rate of the multi-pulses. Our main result is the adiabatic stability of the manifolds subject to this normal hyperbolicity condition. More specifically, the spectrum of the linearization about a fixed N-pulse configuration contains an essential spectrum that is asymptotically close to the origin, as well as semi-strong eigenvalues which move at leading order as the pulse positions evolve. We characterize the semi-strong eigenvalues in terms of the spectrum of an explicit N x N matrix, and rigorously bound the error between the N-pulse manifold and the evolution of the full system, in a polynomially weighted space, so long as the semi-strong spectrum remains strictly in the left-half complex plane, and the essential spectrum is not too close to the origin.

• 出版日期2013