Author
Ghazaryan, Anna Radmir

Year
2005

Advisor
Sandstede, Bjorn

Abstract
We consider a model system, consisting of two nonlinearly coupled partial differential equations, to investigate nonlinear convective instabilities of travelling waves. The system exhibits front solutions which are travelling waves that asymptotically connect two different spatially homogeneous rest states. In the coordinate frame that moves with the front, the rest state ahead of the front is asymptotically stable, while the rest state behind the front experiences a Turing instability: upon increasing a bifurcation parameter, spatially periodic patterns arise. We show that the front is nonlinearly convectively unstable with respect to perturbations that are exponentially localized at positive infinity. More precisely, in the co-moving coordinate frame, the corresponding solution converges pointwise to a translate of the front, and the perturbation to the front is transported toward negative infinity. The proof of this stability result (which appears to be the first of this kind for dissipative systems) is based on the interplay of various norms, including norms for ordinary, uniformly local, and exponentially weighted Sobolev spaces. Among the methods employed are energy and semigroup estimates derived using multiplier theory in uniformly local spaces.

Thesis
Ghazaryan.Anna.Radmir.pdf