361 research outputs found
Time Asymptotics for a Critical Case in Fragmentation and Growth-Fragmentation Equations
Fragmentation and growth-fragmentation equations is a family of problems with
varied and wide applications. This paper is devoted to description of the long
time time asymptotics of two critical cases of these equations, when the
division rate is constant and the growth rate is linear or zero. The study of
these cases may be reduced to the study of the following fragmentation
equation:Using the Mellin transform of the equation, we
determine the long time behavior of the solutions. Our results show in
particular the strong dependence of this asymptotic behavior with respect to
the initial data
Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane
In the present article we consider several issues concerning the doubly
parabolic Keller-Segel system in the plane, when the initial data belong to
critical scaling-invariant Lebesgue spaces. More specifically, we analyze the
global existence of integral solutions, their optimal time decay, uniqueness
and positivity, together with the uniqueness of self-similar solutions. In
particular, we prove that there exist integral solutions of any mass, provided
that \e>0 is sufficiently large. With those results at hand, we are then able
to study the large time behavior of global solutions and prove that in the
absence of the degradation term the solutions behave like self-similar
solutions, while in presence of the degradation term global solutions behave
like the heat kernel
Semi-group theory for the Stokes operator with Navier-type boundary conditions on -spaces
In this article we consider the Stokes problem with Navier-type boundary
conditions on a domain , not necessarily simply connected. Since under
these conditions the Stokes problem has a non trivial kernel, we also study the
solutions lying in the orthogonal of that kernel. We prove the analyticity of
several semigroups generated by the Stokes operator considered in different
functional spaces. We obtain strong, weak and very weak solutions for the time
dependent Stokes problem with the Navier-type boundary condition under
different hypothesis on the initial data and external force
. Then, we study the fractional and pure imaginary powers of
several operators related with our Stokes operators. Using the fractional
powers, we prove maximal regularity results for the homogeneous Stokes problem.
On the other hand, using the boundedness of the pure imaginary powers we deduce
maximal regularity for the inhomogeneous Stokes problem
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