Stability of Singularly Perturbed Solutions to Systems of Reaction-Diffusion Equations

Abstract

Stability theorem is presented for large amplitude singularly perturbed solutions (SPS) of reactiondiffusion systems on a finite interval. Spectral analysis shows that there exists a unique real critical eigenvalue $\lambda _c (\varepsilon )$ which behaves like $\lambda _c (\varepsilon ) \simeq \tau \varepsilon $ as $\varepsilon \downarrow 0$, where $\varepsilon $ is a small parameter contained in the system. All the other noncritical eigenvalues have strictly negative real parts independent of $\varepsilon $. The singular limit eigenvalue problem in §2 plays a key role to judge the sign of $\tau $ which determines the stability of SPS for small $\varepsilon $. Under a natural framework of nonlinearities, $\tau $ becomes negative, namely, SPS is asymptotically stable. Instability result is also shown in §4.