148 research outputs found

### Quantum Zakharov Model in a Bounded Domain

We consider an initial boundary value problem for a quantum version of the
Zakharov system arising in plasma physics. We prove the global well-posedness
of this problem in some Sobolev type classes and study properties of solutions.
This result confirms the conclusion recently made in physical literature
concerning the absence of collapse in the quantum Langmuir waves. In the
dissipative case the existence of a finite dimensional global attractor is
established and regularity properties of this attractor are studied. For this
we use the recently developed method of quasi-stability estimates. In the case
when external loads are $C^\infty$ functions we show that every trajectory from
the attractor is $C^\infty$ both in time and spatial variables. This can be
interpret as the absence of sharp coherent structures in the limiting dynamics.Comment: 27 page

### An intermediate value theorem in ordered Banach spaces

We consider a monotone increasing operator in an ordered Banach space having
$u_-$ and $u_+$ as a strong super- and subsolution, respectively. In contrast
with the well studied case $u_+ < u_-$, we suppose that $u_- < u_+$. Under the
assumption that the order cone is normal and minihedral, we prove the existence
of a fixed point located in the ordered interval $[u_-,u_+].

### Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent

In this paper the long time behaviour of the solutions of 3-D strongly damped
wave equation is studied. It is shown that the semigroup generated by this
equation possesses a global attractor in H_{0}^{1}(\Omega)\times L_{2}(\Omega)
and then it is proved that this global attractor is a bounded subset of
H^{2}(\Omega)\times H^{2}(\Omega) and also a global attractor in
H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)

### Asymptotic behaviour of random tridiagonal Markov chains in biological applications

Discrete-time discrete-state random Markov chains with a tridiagonal
generator are shown to have a random attractor consisting of singleton subsets,
essentially a random path, in the simplex of probability vectors. The proof
uses the Hilbert projection metric and the fact that the linear cocycle
generated by the Markov chain is a uniformly contractive mapping of the
positive cone into itself. The proof does not involve probabilistic properties
of the sample path and is thus equally valid in the nonautonomous deterministic
context of Markov chains with, say, periodically varying transitions
probabilities, in which case the attractor is a periodic path.Comment: 13 pages, 22 bibliography references, submitted to DCDS-B, added
references and minor correction

### The random case of Conley's theorem

The well-known Conley's theorem states that the complement of chain recurrent
set equals the union of all connecting orbits of the flow $\phi$ on the compact
metric space $X$, i.e. $X-\mathcal{CR}(\phi)=\bigcup [B(A)-A]$, where
$\mathcal{CR}(\phi)$ denotes the chain recurrent set of $\phi$, $A$ stands for
an attractor and $B(A)$ is the basin determined by $A$. In this paper we show
that by appropriately selecting the definition of random attractor, in fact we
define a random local attractor to be the $\omega$-limit set of some random
pre-attractor surrounding it, and by considering appropriate measurability, in
fact we also consider the universal $\sigma$-algebra $\mathcal
F^u$-measurability besides $\mathcal F$-measurability, we are able to obtain
the random case of Conley's theorem.Comment: 15 page

### Marketing analysis of the use of drugs containing retinoids in the treatment of acne

Acne is a common skin disease, which is accompanied by a significant skin and psychological burden. Today in Ukraine, significant attention is paid to the issue of acne treatment â€“ scientists from leading scientific, medical (pharmaceutical) and cosmetic institutions are engaged in the search for the most optimal means for the acne treatment. Retinoids are a group of substances that deserves thorough study in terms of production and sustainable introduction of drugs to the pharmaceutical market of Ukraine.
Therefore, the goal of our work was the marketing analysis of drugs for the treatment of acne with the content of retinoids as active substances.
Determining the main trends of foreign manufacturers in the use of retinoids in medicinal products for the acne treatment will allow to predict approaches to the development of domestic effective medicine containing retinoids. During the analysis, methods of logical and meaningful formulation of the problem, office marketing research, content analysis of publications in scientific and practically oriented medical and pharmaceutical publications, comparative analysis, tabular and graphic means of visual presentation of the obtained data were used.
The analysis of the range of drugs containing retinoids, presented on the domestic pharmaceutical market, was carried out according to the data of the State Register of Medicinal Products of Ukraine, the classification system of the ATC, and the State Formulary of Medicinal Products.
According to the results of the work, it was determined that 13 trade names of medicinal products containing retinoids for the acne treatment are registered on the pharmaceutical market of Ukraine. It was determined that there are no registered medicinal products for the acne treatment containing retinoids of domestic production. In Ukraine medicines containing retinoids are represented by six countries. Among the dosage forms used for the acne treatment, solid dosage forms (hard and soft capsules) and soft dosage forms (gels, creams, lotions) should be distinguished. Today, the pharmaceutical market of Ukraine presents drugs of the I, III and IV generation of retinoids. Most of the drugs for the acne treatment containing retinoids registered in Ukraine are monocomponent. The lack of combinations of retinoids with active components widely used for the acne treatment is related to the technological aspects of the production of drugs with retinoids. The volume of sales of drugs with retinoids for the treatment of acne has been increasing in recent years, despite the high cost of products, which indicates the demand for the development of these drugs of domestic production. The obtained data will make it possible to develop approaches to the introduction into the production of domestic drugs for the treatment of acne containing retinoids

### Criteria for strong and weak random attractors

The theory of random attractors has different notions of attraction, amongst
them pullback attraction and weak attraction. We investigate necessary and
sufficient conditions for the existence of pullback attractors as well as of
weak attractors

### Random attractors for degenerate stochastic partial differential equations

We prove the existence of random attractors for a large class of degenerate
stochastic partial differential equations (SPDE) perturbed by joint additive
Wiener noise and real, linear multiplicative Brownian noise, assuming only the
standard assumptions of the variational approach to SPDE with compact
embeddings in the associated Gelfand triple. This allows spatially much rougher
noise than in known results. The approach is based on a construction of
strictly stationary solutions to related strongly monotone SPDE. Applications
include stochastic generalized porous media equations, stochastic generalized
degenerate p-Laplace equations and stochastic reaction diffusion equations. For
perturbed, degenerate p-Laplace equations we prove that the deterministic,
infinite dimensional attractor collapses to a single random point if enough
noise is added.Comment: 34 pages; The final publication is available at
http://link.springer.com/article/10.1007%2Fs10884-013-9294-

### Oscillatory behaviour on a non-autonomous hybrid SIR-Model

We study the impact of some abstract agent intervention on the disease spread modelled by a SIR-model with linear growth infectivity. The intervention is meant to decrease the infectivity, which are activated by a threshold on the number of infected individuals. The coupled model is represented as a nonlinear non-autonomous hybrid system. Stability and reduction results are obtained using the notions of non-autonomous attractors, Bohl exponents, and dichotomy spectrum. Numerical examples are given where the number of infected individuals can oscillate around a equilibrium point or be a succession of bump functions, which are validated with a tool based on the notion of delta-complete decision procedures for solving satisfiability modulo theories problems over the real numbers and bounded delta-reachability. These findings seem to show that hybrid SIR-models are more flexible than standard models and generate a vast set of solution profiles. It also raises questions regarding the possibility of the agent intervention been somehow responsible for the shape and intensity of future outbreaks.publishe

### Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions

A microscopic heterogeneous system under random influence is considered. The
randomness enters the system at physical boundary of small scale obstacles as
well as at the interior of the physical medium. This system is modeled by a
stochastic partial differential equation defined on a domain perforated with
small holes (obstacles or heterogeneities), together with random dynamical
boundary conditions on the boundaries of these small holes.
A homogenized macroscopic model for this microscopic heterogeneous stochastic
system is derived. This homogenized effective model is a new stochastic partial
differential equation defined on a unified domain without small holes, with
static boundary condition only. In fact, the random dynamical boundary
conditions are homogenized out, but the impact of random forces on the small
holes' boundaries is quantified as an extra stochastic term in the homogenized
stochastic partial differential equation. Moreover, the validity of the
homogenized model is justified by showing that the solutions of the microscopic
model converge to those of the effective macroscopic model in probability
distribution, as the size of small holes diminishes to zero.Comment: Communications in Mathematical Physics, to appear, 200

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