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1
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Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise

100%
Jia X., Gao J., Ding X.
Open Mathematics
|
2016
|
tom 14
|
nr 1
586-602
EN
In this paper, we consider the existence of a pullback attractor for the random dynamical system generated by stochastic two-compartment Gray-Scott equation for a multiplicative noise with the homogeneous Neumann boundary condition on a bounded domain of space dimension n ≤ 3. We first show that the stochastic Gray-Scott equation generates a random dynamical system by transforming this stochastic equation into a random one. We also show that the existence of a random attractor for the stochastic equation follows from the conjugation relation between systems. Then, we prove pullback asymptotical compactness of solutions through the uniform estimate on the solutions. Finally, we obtain the existence of a pullback attractor.
2
Content available remote

Global attractivity, oscillation and Hopf bifurcation for a class of diffusive hematopoiesis models

100%
Wang X., Li Z.
Open Mathematics
|
2007
|
tom 5
|
nr 2
397-414
EN
In this paper, we discuss the special diffusive hematopoiesis model $$\frac{{\partial P(t,x)}}{{\partial t}} = \Delta P(t,x) - \gamma P(t,x) + \frac{{\beta P(t - \tau ,x)}}{{1 + P^n (t - \tau ,x)}}$$ with Neumann boundary condition. Sufficient conditions are provided for the global attractivity and oscillation of the equilibrium for Eq. (*), by using a new theorem we stated and proved. When P(t, χ) does not depend on a spatial variable χ ∈ Ω, these results are also true and extend or complement existing results. Finally, existence and stability of the Hopf bifurcation for Eq. (*) are studied.
3
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Decay rates of Volterra equations on ℝN

76%
Conti M., Gatti S., Pata V.
Open Mathematics
|
2007
|
tom 5
|
nr 4
720-732
EN
This note is concerned with the linear Volterra equation of hyperbolic type $$\partial _{tt} u(t) - \alpha \Delta u(t) + \int_0^t {\mu (s)\Delta u(t - s)} ds = 0$$ on the whole space ℝN. New results concerning the decay of the associated energy as time goes to infinity were established.
4
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Blow-up and global existence profile of a class of fully nonlinear degenerate parabolic equations

64%
Li J., Yin J., Jin C.
Open Mathematics
|
2011
|
tom 9
|
nr 6
1435-1447
EN
This paper is mainly concerned with the blow-up and global existence profile for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations with reaction sources.
5
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Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement

64%
Guesmia A.
Nonautonomous Dynamical Systems
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2017
|
tom 4
|
nr 1
78-97
EN
The asymptotic stability of one-dimensional linear Bresse systems under infinite memories was obtained by Guesmia and Kafini [10] (three infinite memories), Guesmia and Kirane [11] (two infinite memories), Guesmia [9] (one infinite memory acting on the longitudinal displacement) and De Lima Santos et al. [6] (one infinite memory acting on the shear angle displacement). When the kernel functions have an exponential decay at infinity, the obtained stability estimates in these papers lead to the exponential stability of the system if the speeds ofwave propagations are the same, and to the polynomial one with decay rate [...] otherwise. The subject of this paper is to study the case where only one infinite memory is considered and it is acting on the vertical displacement. As far as we know, this case has never studied before in the literature. We show that this case is deeply different from the previous ones cited above by proving that the exponential stability does not hold even if the speeds of wave propagations are the same and the kernel function has an exponential decay at infinity. Moreover, we prove that the system is still stable at least polynomially where the decay rate depends on the smoothness of the initial data. For classical solutions, this decay rate is arbitrarily close to [...] . The proof is based on a combination of the energy method and the frequency domain approach to overcome the new mathematical difficulties generated by our system.
6
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Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain

64%
Mel’nyk T. A., Klevtsovskiy A. V.
Open Mathematics
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2017
|
tom 15
|
nr 1
1351-1370
EN
A semi-linear boundary-value problem with nonlinear Robin boundary conditions is considered in a thin 3D aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter 𝓞(ε). Using the multi-scale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε → 0. Namely, we derive the limit problem (ε = 0) in the corresponding graph, define other terms of the asymptotic approximation and prove energetic and uniform pointwise estimates. These estimates allow us to observe the impact of the aneurysm on some properties of the solution.
7
Content available remote

Topological tools for the prescribed scalar curvature problem on S n

64%
Abuzaid D., Ben Mahmoud R., Chtioui H., Rigane A.
Open Mathematics
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2014
|
tom 12
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nr 12
1829-1839
EN
In this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].
8
Content available remote

Global and exponential attractors for a Caginalp type phase-field problem

52%
Bangola B.
Open Mathematics
|
2013
|
tom 11
|
nr 9
1651-1676
EN
We deal with a generalization of the Caginalp phase-field model associated with Neumann boundary conditions. We prove that the problem is well posed, before studying the long time behavior of solutions. We establish the existence of the global attractor, but also of exponential attractors. Finally, we study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist.
9
Content available remote

Global φ-attractor for a modified 3D Bénard system on channel-like domains

52%
Kapustyan O. V., Pankov A. V.
Nonautonomous Dynamical Systems
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2014
|
tom 1
EN
In this paper we prove the existence of a global φ-attractor in the weak topology of the natural phase space for the family of multi-valued processes generated by solutions of a nonautonomous modified 3D Bénard system in unbounded domains for which Poincaré inequality takes place.
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