168 research outputs found
Stable foliations near a traveling front for reaction diffusion systems
We establish the existence of a stable foliation in the vicinity of a
traveling front solution for systems of reaction diffusion equations in one
space dimension that arise in the study of chemical reactions models and solid
fuel combustion. In this way we complement the orbital stability results from
earlier papers by A. Ghazaryan, S. Schecter and Y. Latushkin. The essential
spectrum of the differential operator obtained by linearization at the front
touches the imaginary axis. In spaces with exponential weights, one can shift
the spectrum to the left. We study the nonlinear equation on the intersection
of the unweighted and weighted spaces. Small translations of the front form a
center unstable manifold. For each small translation we prove the existence of
a stable manifold containing the translated front and show that the stable
manifolds foliate a small ball centered at the front
Exponential dichotomies of evolution operators in Banach spaces
This paper considers three dichotomy concepts (exponential dichotomy, uniform
exponential dichotomy and strong exponential dichotomy) in the general context
of non-invertible evolution operators in Banach spaces. Connections between
these concepts are illustrated. Using the notion of Green function, we give
necessary conditions and sufficient ones for strong exponential dichotomy. Some
illustrative examples are presented to prove that the converse of some
implication type theorems are not valid
The essential spectrum of advective equations
A description of the essential spectrum is given for a general class of
linear advective PDE with pseudodifferential bounded perturbation. We prove
that every point in the Sacker-Sell spectrum of the corresponding
bicharacteristic-amplitude system exponentiates into the spectrum of PDE. Exact
spectral pictures are found in various cases. Applications to instability are
presented.Comment: This replaces the earlier version of the paper. The content of the
original submission appeared in two publications -- this present one and the
other one entitled "Cocycles and Ma\~{n}e sequences with an application to
ideal fluids
Stability radius and internal versus external stability in Banach spaces: an evolution semigroup approach
In this paper the theory of evolution semigroups is developed and used to
provide a framework to study the stability of general linear control systems.
These include time-varying systems modeled with unbounded state-space operators
acting on Banach spaces. This approach allows one to apply the classical theory
of strongly continuous semigroups to time-varying systems. In particular, the
complex stability radius may be expressed explicitly in terms of the generator
of a (evolution) semigroup. Examples are given to show that classical formulas
for the stability radius of an autonomous Hilbert-space system fail in more
general settings. Upper and lower bounds on the stability radius are provided
for these general systems. In addition, it is shown that the theory of
evolution semigroups allows for a straightforward operator-theoretic analysis
of internal stability as determined by classical frequency-domain and
input-output operators, even for nonautonomous Banach-space systemsComment: Also at http://www.math.missouri.edu/~stephen/preprint
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