1,443 research outputs found
Spectral analysis of semigroups and growth-fragmentation equations
The aim of this paper is twofold: (1) On the one hand, the paper revisits the
spectral analysis of semigroups in a general Banach space setting. It presents
some new and more general versions, and provides comprehensible proofs, of
classical results such as the spectral mapping theorem, some (quantified)
Weyl's Theorems and the Krein-Rutman Theorem. Motivated by evolution PDE
applications, the results apply to a wide and natural class of generators which
split as a dissipative part plus a more regular part, without assuming any
symmetric structure on the operators nor Hilbert structure on the space, and
give some growth estimates and spectral gap estimates for the associated
semigroup. The approach relies on some factorization and summation arguments
reminiscent of the Dyson-Phillips series in the spirit of those used in
[87,82,48,81]. (2) On the other hand, we present the semigroup spectral
analysis for three important classes of "growth-fragmentation" equations,
namely the cell division equation, the self-similar fragmentation equation and
the McKendrick-Von Foerster age structured population equation. By showing that
these models lie in the class of equations for which our general semigroup
analysis theory applies, we prove the exponential rate of convergence of the
solutions to the associated remarkable profile for a very large and natural
class of fragmentation rates. Our results generalize similar estimates obtained
in \cite{MR2114128,MR2536450} for the cell division model with (almost)
constant total fragmentation rate and in \cite{MR2832638,MR2821681} for the
self-similar fragmentation equation and the cell division equation restricted
to smooth and positive fragmentation rate and total fragmentation rate which
does not increase more rapidly than quadratically. It also improves the
convergence results without rate obtained in \cite{MR2162224,MR2114413} which
have been established under similar assumptions to those made in the present
work
Multiple positive solutions of parabolic systems with nonlinear, nonlocal initial conditions
In this paper we study the existence, localization and multiplicity of
positive solutions for parabolic systems with nonlocal initial conditions. In
order to do this, we extend an abstract theory that was recently developed by
the authors jointly with Radu Precup, related to the existence of fixed points
of nonlinear operators satisfying some upper and lower bounds. Our main tool is
the Granas fixed point index theory. We also provide a non-existence result and
an example to illustrate our theory.Comment: 28 pages, 1 figure. arXiv admin note: text overlap with
arXiv:1401.135
Minimality properties of set-valued processes and their pullback attractors
We discuss the existence of pullback attractors for multivalued dynamical
systems on metric spaces. Such attractors are shown to exist without any
assumptions in terms of continuity of the solution maps, based only on
minimality properties with respect to the notion of pullback attraction. When
invariance is required, a very weak closed graph condition on the solving
operators is assumed. The presentation is complemented with examples and
counterexamples to test the sharpness of the hypotheses involved, including a
reaction-diffusion equation, a discontinuous ordinary differential equation and
an irregular form of the heat equation.Comment: 33 pages. A few typos correcte
On Congruence Compact Monoids
A universal algebra is called congruence compact if every family of congruence classes with the finite intersection property has a non-empty intersection. This paper determines the structure of all right congruence compact monoids S for which Green’s relations J and H coincide. The results are thus sufficiently general to describe, in particular, all congruence compact commutative monoids and all right congruence compact Clifford inverse monoids
On locally compact semitopological -bisimple inverse -semigroups
We describe the structure of Hausdorff locally compact semitopological
-bisimple inverse -semigroups with compact maximal subgroups. In
particular, we show that a Hausdorff locally compact semitopological
-bisimple inverse -semigroup with a compact maximal subgroup is
either compact or topologically isomorphic to the topological sum of its
-classes. We describe the structure of Hausdorff locally compact
semitopological -bisimple inverse -semigroups with a monothetic
maximal subgroups. In particular we prove the dichotomy for locally
compact semitopological Reilly semigroup
with adjoined zero and
with a non-annihilating homomorphism : is
either compact or discrete. At the end we discuss on the remainder under the
closure of the discrete Reilly semigroup
in a semitopological semigroup.Comment: 26 page
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