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

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 au 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 $ au $ which determines the stability of SPS for small $varepsilon $. Under a natural framework of nonlinearities, $ au $ becomes negative, namely, SPS is asymptotically stable. Instability result is also shown in §4.