Using a scalar advection-reaction-diffusion equation with a cubic nonlinearity as a simple model problem, we investigate the effect of domain size on stability and bifurcations of steady states. We focus on two parameter regimes, namely, the regions where the steady state is convectively or absolutely unstable. In the convective-instability regime, the trivial stationary solution is asymptotically stable on any bounded domain but unstable on the real line. To measure the degree to which the trivial solution is stable, we estimate the distance of the trivial solution to the boundary of its basin of attraction: We show that this distance is exponentially small in the diameter of the domain for subcritical nonlinearities, while it is bounded away from zero uniformly in the domain size for supercritical nonlinearities. Lastly, at the onset of the absolute instability where the trivial steady state destabilizes on large bounded domains, we discuss bifurcations and amplitude scalings.
Bibliographical noteFunding Information:
B. Sandstede was partially supported by the Alfred P Sloan Foundation and by the NSF through Grant DMS-0203854. A. Sc heel was partially supported by the NSF through Grant DMS-0203301. We acknowledge helpful discussions with Peter Polacik.
- Absolute instability
- Basin of attraction
- Convective instability