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1.
Pullback exponential attractors for nonautonomous dynamical system in space of higher regularity 下载免费PDF全文
Yongjun Li Tinggang Zhao Hongqing Wu Jinying Wei 《Journal of Applied Analysis & Computation》2016,6(1):242-253
Under what condition, a process which exists a $(E,E)$-pullback exponential attractor implies the existence of $(E,V)$- pullback exponential attractor when $V$ embedded in $E$? We answer this question in this paper. As an application of this result, we prove the existence of pullback exponential attractor for a nonlinear reaction-diffusion equation with a polynomial growth nonlinearity in $L^q(\Omega)(\forall q\geq 2)$ and $H_0^1(\Omega)$. 相似文献
2.
Dalibor Pražák 《Mathematical Methods in the Applied Sciences》2007,30(17):2197-2214
We study the dynamics of an incompressible, homogeneous fluid of a power‐law type, with the stress tensor T = ν(1 + µ|Dv|)p?2Dv, where Dv is a symmetric velocity gradient. We consider the two‐dimensional problem with periodic boundary conditions and p ∈ (1, 2). Under these assumptions, we estimate the fractal dimension of the exponential attractor, using the so‐called method of ??‐trajectories. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
3.
《Mathematical Methods in the Applied Sciences》2018,41(1):234-249
This paper is concerned with the two‐species chemotaxis‐competition system where Ω is a bounded domain in with smooth boundary ∂Ω, n≥2; χi and μi are constants satisfying some conditions. The above system was studied in the cases that a1,a2∈(0,1) and a1>1>a2, and it was proved that global existence and asymptotic stability hold when are small. However, the conditions in the above 2 cases strongly depend on a1,a2, and have not been obtained in the case that a1,a2≥1. Moreover, convergence rates in the cases that a1,a2∈(0,1) and a1>1>a2 have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all a1,a2>0 which covers the case that a1,a2≥1, and lead to convergence rates for solutions of the above system in the cases that a1,a2∈(0,1) and a1≥1>a2. 相似文献
4.
Gianluca Mola 《Mathematical Methods in the Applied Sciences》2009,32(18):2368-2404
We consider a conserved phase‐field system on a tri‐dimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ?, which is represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore, the system consists of a linear integrodifferential equation for ?, which is coupled with a viscous Cahn–Hilliard type equation governing the order parameter χ. The latter equation contains a nonmonotone nonlinearity ? and the viscosity effects are taken into account by a term ?αΔ?tχ, for some α?0. Rescaling the kernel k with a relaxation time ε>0, we formulate a Cauchy–Neumann problem depending on ε and α. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {?α,ε} for our problem, whose basin of attraction can be extended to the whole phase–space in the viscous case (i.e. when α>0). Moreover, we prove that the symmetric Hausdorff distance of ?α,ε from a proper lifting of ?α,0 tends to 0 in an explicitly controlled way, for any fixed α?0. In addition, the upper semicontinuity of the family of global attractors {??α,ε} as ε→0 is achieved for any fixed α>0. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
5.
《Mathematical Methods in the Applied Sciences》2018,41(8):3138-3154
This paper considers the 2‐species chemotaxis‐Stokes system with competitive kinetics under homogeneous Neumann boundary conditions in a 3‐dimensional bounded domain with smooth boundary. Both chemotaxis‐fluid systems and 2‐species chemotaxis systems with competitive terms were studied by many mathematicians. However, there have not been rich results on coupled 2‐species–fluid systems. Recently, global existence and asymptotic stability in the above problem with (u·∇)u in the fluid equation were established in the 2‐dimensional case. The purpose of this paper is to give results for global existence, boundedness, and stabilization of solutions to the above system in the 3‐dimensional case when is sufficiently small. 相似文献
6.
Stefania Gatti Maurizio Grasselli Alain Miranville Vittorino Pata 《Proceedings of the American Mathematical Society》2006,134(1):117-127
Given a dissipative strongly continuous semigroup depending on some parameters, we construct a family of exponential attractors which is robust, in the sense of the symmetric Hausdorff distance, with respect to (even singular) perturbations.
7.
In this paper, we consider a derivative Ginzburg-Landau-type equation with periodic initial-value condition in three-dimensional spaces. Sufficient conditions for existence and uniqueness of a global solution are obtained by uniform a priori estimates of the solution. Furthermore, the existence of a global attractor and an exponential attractor with finite dimensions are proved. 相似文献
8.
Liangchen Wang Chunlai Mu Xuegang Hu Ya Tian 《Mathematical Methods in the Applied Sciences》2017,40(8):3000-3016
This paper studies the chemotaxis‐haptotaxis system with nonlinear diffusion subject to the homogeneous Neumann boundary conditions and suitable initial conditions, where χ , ξ and μ are positive constants, and (n ?2) is a bounded and smooth domain. Here, we assume that D (u )?c D u m ? 1 for all u > 0 with some c D > 0 and m ?1. For the case of non‐degenerate diffusion, if μ > μ ?, where it is proved that the system possesses global classical solutions which are uniformly‐in‐time bounded. In the case of degenerate diffusion, we show that the system admits a global bounded weak solution under the same assumptions. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
9.
Zhou Shengfan 《Proceedings of the American Mathematical Society》1999,127(12):3623-3631
An estimate on the Hausdorff dimension of the global attractor for damped nonlinear wave equations, in two cases of nonlinear damping and linear damping, with Dirichlet boundary condition is obtained. The gained Hausdorff dimension is bounded and is independent of the concrete form of nonlinear damping term. In the case of linear damping, the gained Hausdorff dimension remains small for large damping, which conforms to the physical intuition.
10.
Alain Miranville 《Central European Journal of Mathematics》2006,4(1):163-182
Our aim in this paper is to study the long time behavior of a class of doubly nonlinear parabolic equations. In particular,
we prove the existence of the global attractor which has, in one and two space dimensions, finite fractal dimension. 相似文献
11.
Rui-feng Zhang 《应用数学学报(英文版)》2008,24(1):19-28
The long time behavior of solutions of the generalized Hasegawa-Mima equation with dissipation term is considered. The existence of global attractors of the periodic initial value problem is proved, and the estimate of the upper bound of the Hausdorff and fractal dimensions for the global attractors is obtained by means of uniform a priori estimates method. 相似文献
12.
《Mathematical Methods in the Applied Sciences》2018,41(10):3790-3819
In this paper, we construct the pullback exponential attractors for evolution processes in which the difference of 2 solutions lacks the smoothing property. To do this, by the uniform squeezing property of the corresponding discrete process, we add the points to the pullback attractor such that every new set of it has the finite fractal dimension and pullback exponentially attracts every bounded subset of the phase space. As the applications, we establish the existence of pullback exponential attractors for non‐autonomous reaction‐diffusion equation without any restriction on the growing order of nonlinear term and non‐autonomous strongly damped wave equation in with critical nonlinearity. 相似文献
13.
We consider the semilinear reaction diffusion equation
, in a bounded domain . We assume the standard “Allen-Cahn-type” nonlinearity, while V is either the inverse square potential or the borderline potential (thus including the classical Allen-Cahn-type equation as a special case when ). In the subcritical cases and where is the optimal constant of Hardy and Hardy-type inequalities), we present a new estimate on the dimension of the global attractor.
This estimate comes out by an improved lower bound for sums of eigenvalues of the Laplacian by A. D. Melas (Proc. Amer. Math.
Soc. 131 (2003), 631–636). The estimate is sharp, revealing the existence of (an explicitly given) threshold value for the ratio of
the volume to the moment of inertia of Ω on which the dimension of the attractor may considerably change. Consideration is
also given on the finite dimensionality of the global attractor in the critical case
Received: 7 May 2008 相似文献
14.
We consider the following doubly nonlinear parabolic equation in a bounded domain Ω??3: where the nonlinearity f is allowed to have a degeneracy with respect to ?tu of the form ?tu|?tu|p at some points x∈Ω. Under some natural assumptions on the nonlinearities f and g, we prove the existence and uniqueness of a solution of that problem and establish the finite‐dimensionality of global and exponential attractors of the semigroup associated with this equation in the appropriate phase space. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
15.
Uniform regularity and vanishing viscosity limit for the chemotaxis‐Navier‐Stokes system in a 3D bounded domain 下载免费PDF全文
Zhipeng Zhang 《Mathematical Methods in the Applied Sciences》2017,40(18):7564-7597
We investigate the uniform regularity and vanishing viscosity limit for the incompressible chemotaxis‐Navier‐Stokes system with Navier boundary condition for velocity field and Neumann boundary condition for cell density and chemical concentration in a 3D bounded domain. It is shown that there exists a unique strong solution of the incompressible chemotaxis‐Navier‐Stokes system in a finite time interval, which is independent of the viscosity coefficient. Moreover, this solution is uniformly bounded in a conormal Sobolev space, which allows us to take the vanishing viscosity limit to obtain the incompressible chemotaxis‐Euler system. 相似文献
16.
Messoud Efendiev 《Journal of Mathematical Analysis and Applications》2009,358(1):136-147
In this paper we continue systematic study of the dimension estimate of the global attractors for the chemotaxis-growth system and its finite-element approximations. Utilizing a conservative upwind finite-element scheme we managed significantly to improve dimension estimates with respect to the chemotactic parameter. 相似文献
17.
In this paper we study well‐posedness of the damped nonlinear wave equation in Ω × (0, ∞) with initial and Dirichlet boundary condition, where Ω is a bounded domain in ?2; ω?0, ωλ1+µ>0 with λ1 being the first eigenvalue of ?Δ under zero boundary condition. Under the assumptions that g(·) is a function with exponential growth at the infinity and the initial data lie in some suitable sets we establish several results concerning local existence, global existence, uniqueness and finite time blow‐up property and uniform decay estimates of the energy. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
18.
Messoud Efendiev 《Mathematical Methods in the Applied Sciences》2011,34(8):939-949
We study in this article the long‐time behavior of solutions of fourth‐order parabolic equations in bfR3. In particular, we prove that under appropriate assumptions on the nonlinear interaction function and on the external forces, these equations possess infinite‐dimensional exponential attractors whose Kolmogorov's ε‐entropy satisfies an estimate of the same type as that obtained previously for the ε‐entropy of the global attractor. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
19.
Finite‐dimensional global attractor for a semi‐discrete fractional nonlinear Schrödinger equation 下载免费PDF全文
Caterina Calgaro Olivier Goubet Ezzeddine Zahrouni 《Mathematical Methods in the Applied Sciences》2017,40(15):5563-5574
We consider a semi‐discrete in time Crank–Nicolson scheme to discretize a weakly damped forced nonlinear fractional Schrödinger equation u t ?i (?Δ)α u +i |u |2u +γ u =f for considered in the the whole space . We prove that such semi‐discrete equation provides a discrete infinite‐dimensional dynamical system in that possesses a global attractor in . We show also that if the external force is in a suitable weighted Lebesgue space, then this global attractor has a finite fractal dimension. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
20.
Xie Li 《Mathematical Methods in the Applied Sciences》2016,39(2):289-301
This paper is devoted to the attraction–repulsion chemotaxis system with nonlinear diffusion: where χ > 0, ζ > 0, αi>0, βi>0 (i = 1,2) and f(s)≤κ ? μsτ. In two‐space dimension, we prove the global existence and uniform boundedness of the classical solution to this model for any μ > 0. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献