66 research outputs found
Harmonic solutions to a class of differential-algebraic equations with separated variables
We study the set of T-periodic solutions of a class of T-periodically
perturbed Differential-Algebraic Equations with separated variables. Under
suitable hypotheses, these equations are equivalent to separated variables ODEs
on a manifold. By combining known results on Differential-Algebraic Equations,
with an argument about ODEs on manifolds, we obtain a global continuation
result for the T-periodic solutions to the considered equations. As an
application of our method, a multiplicity result is provided
About the notion of non--resonance and applications to topological multiplicity results for ODEs on differentiable manifolds
By using topological methods, mainly the degree of a tangent vector field, we
establish multiplicity results for -periodic solutions of parametrized
-periodic perturbations of autonomous ODEs on a differentiable manifold .
In order to provide insights into the key notion of -resonance, we consider
the elementary situations and . So doing, we
provide more comprehensive analysis of those cases and find improved
conditions.Comment: 10 figure
On a class of differential-algebraic equations with infinite delay
We study the set of -periodic solutions of a class of -periodically
perturbed Differential-Algebraic Equations, allowing the perturbation to
contain a distributed and possibly infinite delay. Under suitable assumptions,
the perturbed equations are equivalent to Retarded Functional (Ordinary)
Differential Equations on a manifold. Our study is based on known results about
the latter class of equations.Comment: 13 pages. Revision: Incorporate changes suggested by readers.
Corrected a few typos across the paper, definition of BU added, revised the
(previously incorrect) definition of solution of RFDAE, made slight changes
in the Introduction. Replacement of Dec. 6, 2012: introduced further changes
suggested by referee, bundled addendum/erratum containing a corrected version
of Lemma 5.5 and Corollary 5.
On a non-homogeneous and non-linear heat equation
We consider the Cauchy-problem for a parabolic equation of the following
type:
\begin{equation*}
\frac{\partial u}{\partial t}= \Delta u+ f(u,|x|),
\end{equation*} where is supercritical. We supply this equation
by the initial condition , and we allow to be either
bounded or unbounded in the origin but smaller than stationary singular
solutions. We discuss local existence and long time behaviour for the solutions
for a wide class of non-homogeneous non-linearities . We show
that in the supercritical case, Ground States with slow decay lie on the
threshold between blowing up initial data and the basin of attraction of the
null solution. Our results extend previous ones allowing Matukuma-type
potential and more generic dependence on .
Then, we further explore such a threshold in the subcritical case too. We
find two families of initial data and which are
respectively above and below the threshold, and have arbitrarily small distance
in norm, whose existence is new even for . Quite
surprisingly both and have fast decay (i.e. ), while the expected critical asymptotic behavior is slow decay
(i.e. ).Comment: 2 figure
Periodic solutions of semi-explicit differential-algebraic equations with time-dependent constraints
In this paper we investigate the properties of the set of T-periodic
solutions of semi-explicit parametrized Differential-Algebraic Equations with
non-autonomous constraints of a particular type. We provide simple, degree
theoretic conditions for the existence of branches of T-periodic solutions of
the considered equations. Our approach is based on topological arguments about
differential equations on implicitly defined manifolds, combined with
elementary facts of matrix analysis
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