Professor Sergey Zelik

We report on new results concerning the global well-posedness, dissipativity
and attractors of the damped quintic wave equations in bounded domains of R^3.

We study initial boundary value problems for the convective Cahn-Hilliard
equation $\Dt u +\px^4u +u\px u+\px^2(|u|^pu)=0$. It is well-known that without
the convective term, the solutions of this equation may blow up in finite time
for any $p>0$. In contrast to that, we show that the presence of the convective
term $u\px u$ in the Cahn-Hilliard equation prevents blow up at least for
$0

large enough ($p\ge2$). The related equations like
Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard
equation, are also considered.

The paper is devoted to a rigorous construction of a parabolic system of partial differential equations which displays space-time chaotic behavior in its global attractor. The construction starts from a periodic array of identical copies of a temporally chaotic reaction-diffusion system (RDS) on a bounded domain with Dirichlet boundary conditions. We start with the case without coupling where space-time chaos, defined via embedding of multi- dimensional Bernoulli schemes, is easily obtained. We introduce small coupling by replacing the Dirichlet boundary conditions by strong absorption between the active islands. Using hyperbolicity and delicate PDE estimates we prove persistence of the embedded Bernoulli scheme. Furthermore we smoothen the nonlinearity and obtain a RDS which has polynomial interaction terms with space and time-periodic coefficients and which has a hyperbolic invariant set on which the dynamics displays spatio-temporal chaos. Finally we show that such a system can be embedded in a bigger system which is autonomous and homogeneous and still contains space-time chaos. Obviously, hyperbolicity is lost in this step. © Springer Science+Business Media, LLC 2007.

The paper is devoted to constructing a random exponential attractor for some
classes of stochastic PDE's. We first prove the existence of an exponential
attractor for abstract random dynamical systems and study its dependence on a
parameter and then apply these results to a nonlinear reaction-diffusion system
with a random perturbation. We show, in particular, that the attractors can be
constructed in such a way that the symmetric distance between the attractors
for stochastic and deterministic problems goes to zero with the amplitude of
the random perturbation.

In this article, we construct robust (i.e. lower and upper semicontinuous) exponential attractors for singularly perturbed phase-field type equations. Moreover, we obtain estimates for the symmetric distance between these exponential attractors and that of the limit Cahn-Hilliard equation in terms of the perturbation parameter. We can note that the continuity is obtained without time shifts as it is the case in previous results. © 2002 Southwest Texas State University.

The long-time behaviour of bounded solutions of a reaction-diffusion system in an unbounded domain © ‚ ?n, for which the nonlinearity f(u, ?xu) explicitly depends on ?xu is studied. We prove the existence of a global attractor, fractal dimension of which is infinite, and give upper and lower bounds for the Kolmogorov entropy of the attractor and analyze the sharpness of these bounds. © 2002 Plenum Publishing Corporation.

A strongly damped wave equation including the displacement depending
nonlinear damping term and nonlinear interaction function is considered. The
main aim of the note is to show that under the standard dissipativity
restrictions on the nonlinearities involved the initial boundary value problem
for the considered equation is globally well-posed in the class of sufficiently
regular solutions and the semigroup generated by the problem possesses a global
attractor in the corresponding phase space. These results are obtained for the
nonlinearities of an arbitrary polynomial growth and without the assumption
that the considered problem has a global Lyapunov function.

In this paper the quasi-linear second-order parabolic systems of reaction-diffusion type in an unbounded domain are considered. Our aim is to study the long-time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors on the weight of the Sobolev spaces is considered. © 2001 John Wiley & Sons, Inc.

A system consisting of a chaotic (billiard-like) oscillator coupled to a
linear wave equation in the three-dimensional space is considered. It is shown
that the chaotic behaviour of the oscillator can cause the transfer of energy
from a monochromatic wave to the oscillator, whose energy can grow without
bound.

We consider interpolation inequalities for imbeddings of the l
2-sequence
spaces over d-dimensional lattices into the l
?
0 spaces written as interpolation inequality
between the l
2-norm of a sequence and its difference. A general method is developed
for finding sharp constants, extremal elements and correction terms in this type of
inequalities. Applications to Carlson?s inequalities and spectral theory of discrete
operators are given.

We study the system of Euler equations with the so-called Ekman damping in the whole 2D space. The global well-posedness and dissipativity for the weak infinite energy solutions of this problem in the uniformly local spaces is verified based on the further development of the weighted energy theory for the Navier?Stokes and Euler type problems. In addition, the existence of weak locally compact global attractor is proved and some extra compactness of this attractor is obtained.

The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using a proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.

A nonlinear parabolic equation of the fourth order is analyzed. The equation is characterized by a mobility coefficient that degenerates at 0. Existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a priori estimates. If a viscosity term is added and additional conditions on the nonlinear terms are assumed, then it is proved that any weak solution becomes instantaneously strictly positive. This in particular implies uniqueness for strictly positive times and further time-regularization properties. The long-time behavior of the problem is also investigated and the existence of trajectory attractors and, under more restrictive conditions, of strong global attractors is shown. © 2012 American Mathematical Society.

We study the infinite-energy solutions of the Cahn-Hilliard equation in the whole 3D space in uniformly local phase spaces. In particular, we establish the global existence of solutions for the case of regular potentials of arbitrary polynomial growth and for the case of sufficiently strong singular potentials. For these cases, the uniqueness and further regularity of the obtained solutions are proved as well. We discuss also the analogous problems for the case of the so-called Cahn-Hilliard-Oono equation where, in addition, the dissipativity of the associated solution semigroup is established.

In this paper we analyze a nonlinear parabolic equation characterized by a
singular diffusion term describing very fast diffusion effects. The equation is
settled in a smooth bounded three-dimensional domain and complemented with a
general boundary condition of dynamic type. This type of condition prescribes
some kind of mass conservation; hence extinction effects are not expected for
solutions that emanate from strictly positive initial data. Our main results
regard existence of weak solutions, instantaneous regularization properties,
long-time behavior, and, under special conditions, uniqueness.

We study the so-called damped Navier-Stokes equations in the whole 2D space. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces are verified based on the further development of the weighted energy theory for the Navier-Stokes type problems. Note that any divergent free vector field u0 L? R2) is allowed and no assumptions on the spatial decay of solutions as |x| to ? are posed. In addition, applying the developed theory to the case of the classical Navier-Stokes problem in R2, we show that the properly defined weak solution can grow at most polynomially (as a quintic polynomial) as time goes to infinity. © 2013 Springer Basel.

A semilinear strongly damped wave equation with memory is considered in the past history framework. The existence of global attractors of optimal regularity is established, both for critical and supercritical nonlinearities, under a necessary and sufficient condition on the memory kernel.

Galenko et al. proposed a modified Cahn-Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This equation contains an inertial term which causes the loss of any regularizing effect on the solutions. Here we consider an initial and boundary value problem for this equation in a two-dimensional bounded domain. We prove a number of results related to well-posedness and large time behavior of solutions. In particular, we analyze the existence of bounded absorbing sets in two different phase spaces and, correspondingly, we establish the existence of the global attractor. We also demonstrate the existence of an exponential attractor.

We give an analytic proof of the existence of Shilnikov chaos in complex
Ginzburg-Landau equation subject to a large third-order dispersion
perturbation.

We consider in this article a nonlinear reaction-diffusion system with a transport term (L,? x )u, where L is a given vector field, in an unbounded domain ©. We prove that, under natural assumptions, this system possesses a locally compact attractor script A sign in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov's µ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in © . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584-637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on script A sign. As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of script A sign. © Springer Science+Business Media, LLC 2006.

We study the asymptotic behavior of solutions of one coupled PDE-ODE system
arising in mathematical biology as a model for the development of a forest
ecosystem.
In the case where the ODE-component of the system is monotone, we establish
the existence of a smooth global attractor of finite Hausdorff and fractal
dimension.
The case of the non-monotone ODE-component is much more complicated. In
particular, the set of equilibria becomes non-compact here and contains a huge
number of essentially discontinuous solutions. Nevertheless, we prove the
stabilization of any trajectory to a single equilibrium if the coupling
constant is small enough.

A general method is proposed for finding sharp constants for the embeddings of the Sobolev spaces H (M) on an n-dimensional Riemannian manifold M into the space of bounded continuous functions, where m > n/2. The method is based on an analysis of the asymptotics with respect to the spectral parameter of the Green's function of an elliptic operator of order 2m whose square root has domain determining the norm of the corresponding Sobolev space. The cases of the n-dimensional torus T and the n-dimensional sphere S are treated in detail, as well as certain manifolds with boundary. In certain cases when M is compact, multiplicative inequalities with remainder terms of various types are obtained. Inequalities with correction terms for periodic functions imply an improvement for the well-known Carlson inequalities. © 2014 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with a nonlinearity of arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also considered.

The nonlinear reaction-diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor A in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's µ-entropy. Upper and lower bounds of this entropy are obtained. Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ?n. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional "time" if n > 1) acting on a phase space A. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy. In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz-continuous) embedding of this system into the spatial shifts on the attractor. Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.

The dissipative wave equation with a critical quintic nonlinearity in smooth
bounded three dimensional domain is considered. Based on the recent extension
of the Strichartz estimates to the case of bounded domains, the existence of a
compact global attractor for the solution semigroup of this equation is
established. Moreover, the smoothness of the obtained attractor is also shown.

We give a comprehensive study of strong uniform attractors of non-autonomous
dissipative systems for the case where the external forces are not translation
compact. We introduce several new classes of external forces which are not
translation compact, but nevertheless allow to verify the attraction in a
strong topology of the phase space and discuss in a more detailed way the class
of so-called normal external forces introduced before. We also develop a
unified approach to verify the asymptotic compactness for such systems based on
the energy method and apply it to a number of equations of mathematical physics
including the Navier-Stokes equations, damped wave equations and
reaction-diffusing equations in unbounded domains.

We consider interpolation inequalities for imbeddings of the $l^2$-sequence
spaces over $d$-dimensional lattices into the $l^\infty_0$ spaces written as
interpolation inequality between the $l^2$-norm of a sequence and its
difference. A general method is developed for finding sharp constants, extremal
elements and correction terms in this type of inequalities. Applications to
Carlson's inequalities and spectral theory of discrete operators are given.

We give a comprehensive study of interpolation inequalities for periodic
functions with zero mean, including the existence of and the asymptotic
expansions for the extremals, best constants, various remainder terms, etc.
Most attention is paid to the critical (logarithmic) Sobolev inequality in the
two-dimensional case, although a number of results concerning the best
constants in the algebraic case and different space dimensions are also
obtained.

© 2014, Springer Science+Business Media New York.We develop an efficient and robust numerical scheme to compute multi-fronts in one-dimensional real Ginzburg?Landau equations that range from well-separated to strongly interacting and colliding. The scheme is based on the global centre-manifold reduction where one considers an initial sum of fronts plus a remainder function (not necessarily small) and applying a suitable projection based on the neutral eigenmodes of each front. Such a scheme efficiently captures the weakly interacting tails of the fronts. Furthermore, as the fronts become strongly interacting, we show how they may be added to the remainder function to accurately compute through collisions. We then present results of our numerical scheme applied to various real Ginzburg Landau equations where we observe colliding fronts, travelling fronts and fronts converging to bound states. Finally, we discuss how this numerical scheme can be extended to general PDE systems and other multi-localised structures.

In this presentation, interaction of stationary and pulsating localized structures of light in active and passive optical devices is studied analytically and numerically. Being close enough to each other, optical pulses interact via decaying tails. Interference between the tails can produce intensity oscillations responsible for the formation of pulse bound states. Using an asymptotic approach we derive and analyze a set of ordinary differential equations governing the slow time evolution of the parameters of individual pulses, such as their coordinates, optical and oscillation phases. Being independent of specific details of the model, the form of these "interaction equations" is determined mainly by the asymptotic behavior of the pulse tails and the symmetries of the model equations. They have a universal nature and can be used to study interaction of temporal or spatial localized structures not only in optical, but also in hydrodynamic, plasma, and even biological systems.

Dissipative wave equations with critical quintic nonlinearity and damping
term involving the fractional Laplacian are considered. The additional
regularity of energy solutions is established by constructing the new
Lyapunov-type functional and based on this, the global well-posedness and
dissipativity of the energy solutions as well as the existence of a smooth
global and exponential attractors of finite Hausdorff and fractal dimension is
verified.

Our aim in this article is to discuss recent issues related with the Cahn-
Hilliard equation in phase separation with the thermodynamically relevant logarithmic
potentials; in particular, we are interested in the well-posedness and
the study of the asymptotic behavior of the solutions (and, more precisely, the
existence of finite-dimensional attractors). We first consider the usual Neumann
boundary conditions and then dynamic boundary conditions which account for
the interactions with the walls in confined systems and have recently been proposed
by physicists. We also present, in the case of dynamic boundary conditions,
some numerical results.

A new type of dynamic stable bound state of dissipative localized structures is found. It is characterized by chaotic oscillations of distance between the localized structures, their phase difference, and the center of mass velocity. © 2005 Optical Society of America.

This paper is a study of global attractors of abstract semilinear parabolic equations and their embeddings in finite-dimensional manifolds. As is well known, a sufficient condition for the existence of smooth (at least C 1-smooth) finite-dimensional inertial manifolds containing a global attractor is the so-called spectral gap condition for the corresponding linear operator. There are also a number of examples showing that if there is no gap in the spectrum, then a C1-smooth inertial manifold may not exist. On the other hand, since an attractor usually has finite fractal dimension, by Mañé's theorem it projects bijectively and Hölder- homeomorphically into a finite-dimensional generic plane if its dimension is large enough. It is shown here that if there are no gaps in the spectrum, then there exist attractors that cannot be embedded in any Lipschitz or even log-Lipschitz finite-dimensional manifold. Thus, if there are no gaps in the spectrum, then in the general case the inverse Mañé projection of the attractor cannot be expected to be Lipschitz or log-Lipschitz. Furthermore, examples of attractors with finite Hausdorff and infinite fractal dimension are constructed in the class of non-linearities of finite smoothness. Bibliography: 35 titles. © 2013 RAS(DoM) and LMS.

An inertial manifold for the system of 1D reaction-diffusion-advection
equations endowed by the Dirichlet boundary conditions is constructed. Although
this problem does not initially possess the spectral gap property, it is shown
that this property is satisfied after the proper non-local change of the
dependent variable.

In the half-cylinder ©+ = ?+ × w, w ? ?n, we study a second-order system of elliptic equations containing a non-linear function f(u, cursive Greek chi0, cursive Greek chi2) = (f1,ï, fk) and right-hand side g(cursive Greek chi0, cursive Greek chi2) = (g1,ï, gk), cursive Greek chi0 ? ?+, cursive Greek chi2 ? w. If these functions satisfy certain conditions, then it is proved that the first boundary-value problem for this system has at least one solution belonging to the space [Hloc2,p(©+)]k , p > n + 1. We study the behaviour of the solutions u(cursive Greek chi0, cursive Greek chi2) of this system as cursive Greek chi0 ’ +?. Along with the original system we study the family of systems obtained from it through shifting with respect to cursive Greek chi0 by all h, h e 0. A semigroup {T(h), h e 0}, T(h)u(cursive Greek chi0, ·) = u(cursive Greek chi0 + h, ·) acts on the set of solutions š+ of these systems of equations. It is proved that this semigroup has a trajectory attractor double-struck A consisting of the solutions v(cursive Greek chi0, cursive Greek chi2) in š+ that admit a bounded extension to the entire cylinder © = ? × w. Solutions u(cursive Greek chi0, cursive Greek chi2) ? š+ are attracted by the attractor double-struck A as cursive Greek chi0 ’ +?. We give a number of applications and consider some questions of the theory of perturbations of the original system of equations.

The longtime behaviour of solutions of a reaction-diffusion system with the nonlinearity rapidly oscillating in time (f = f(t/µ, u)) is studied. It is proved that (under the natural assumptions) this behaviour can be described in terms of global (uniform) attractors Aµ of the corresponding dynamical process and that these attractors tend as µ ’ 0 to the global attractor A0 of the averaged autonomous system. Moreover, we give a detailed description of the attractors Aµ, µ ? 1, in the case where the averaged system possesses a global Liapunov function.

We study regular global attractors of dissipative dynamical semigroups with discrete or continuous time and we investigate attractors for nonautonomous perturbations of such semigroups. The main theorem states that the regularity of global attractors is preserved under small nonautonomous perturbations. Moreover, nonautonomous regular global attractors remain exponential and robust. We apply these general results to model nonautonomous reaction-diffusion systems in a bounded domain of ? with time-dependent external forces. © 2013 RAS(DoM) and LMS.

This paper is devoted to the problem of finite-dimensional reduction for parabolic partial differential equations. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mañé projection theorems. The recent counter-examples showing that the underlying dynamics may in a sense be infinite dimensional if the spectral gap condition is violated, as well as a discussion of the most important open problems, are also included.

We study the impact of the convective terms on the global solvability or
finite time blow up of solutions of dissipative PDEs. We consider the model examples of 1D Burger's type equations, convective Cahn-Hilliard equation, generalized Kuramoto-Sivashinsky equation and KdV type equations, we establish the following common scenario: adding sufficiently strong (in comparison with the destabilizing nonlinearity) convective terms to equation prevents the solutions from blowing up in finite time and makes the considered system globally well-posed and dissipative and for weak enough convective terms the finite time blow up may occur similarly to the case when the equation does not involve convective term. This kind of result has been previously known for the case of Burger's type equations and has been strongly based on maximum principle. In contrast to this, our results are based on the weighted energy estimates which do not require the maximum principle for the considered problem.

We study a Penrose-Fife phase transition model coupled with homogeneous
Neumann boundary conditions. Improving previous results, we show that the
initial value problem for this model admits a unique solution under weak
conditions on the initial data. Moreover, we prove asymptotic regularization
properties of weak solutions.

The behavior as t ’ ? of solutions of the equation ? t2u + ³? tu + Au + f(t)u = 0 in a Hilbert space is studied, where A = A* is a positive definite operator with compact inverse and the operator f is periodic in t. The notion of instability index is introduced for this equation; we prove that the instability index is finite under natural assumptions (f must be dominated by A). Asymptotic estimates of the instability index are obtained as ³ ’ 0, and an example is constructed showing that they cannot be improved. Furthermore, we study the qualitative characteristics of the spectrum of the monodromy operator and the existence of the Floquet representation for this problem. ©1997 Plenum Publishing Corporation.

We give a detailed study of the infinite-energy solutions of the Cahn-Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well-posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed to prove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and study its properties.

An inertial manifold (IM) is one of the key objects in the modern theory of dissipative systems generated by partial differential equations (PDEs) since it allows us to describe the limit dynamics of the considered system by the reduced finite-dimensional system of ordinary differential equations (ODEs). It is well known that the existence of an IM is guaranteed when the so called spectral gap conditions are satisfied, whereas their violation leads to the possibility of an infinite-dimensional limit dynamics, at least on the level of an abstract parabolic equation. However, these conditions restrict greatly the class of possible applications and are usually satisfied in the case of one spatial dimension only.

Despite many efforts in this direction, the IMs in the case when the spectral gap conditions are violated remain a mystery especially in the case of parabolic PDEs. On the one hand, there is a number of interesting classes of such equations where the existence of IMs is established without the validity of the spectral gap conditions and, on the other hand there were no examples of dissipative parabolic PDEs where the non-existence of an IM is rigorously proved.

The main aim of this thesis is to bring some light on this mystery by the comprehensive study of three model examples of parabolic PDEs where the spectral gap conditions are not satisfied, namely, 1D reaction-diffusion-advection (RDA) systems (see Chapter 3), the 3D Cahn-Hilliard equation on a torus (see Chapter 4) and the modified 3D Navier-Stokes equations (see Chapter 5). For all these examples the existence or non-existence of IM was an open problem.

As shown in Chapter 3, the existence or non-existence of an IM for RDA systems strongly depends on the boundary conditions. In the case of Dirichlet or Neumann boundary conditions, we have proved the existence of an IM using a specially designed non-local in space diffeomorphism which transforms the equations to the new ones for which the spectral gap conditions are satisfied. In contrast to this, in the case of periodic boundary conditions, we construct a natural example of a RDA system which does not possess an IM.

In Chapters 4 and 5 we develop an extension of the so-called spatial averaging principle (SAP) (which has been suggested by Sell and Mallet-Paret in order to treat scalar reaction-diffusion equation on a 3D torus) to the case of 4th order equations where the nonlinearity loses smoothness (the Cahn-Hilliard equation) as well as

The effect of rapid oscillations, related to large dispersion terms, on the dynamics of dissipative evolution equations is studied for the model examples of the 1D complex Ginzburg-Landau and the Kuramoto-Sivashinsky equations. Three different scenarios of this effect are demonstrated. According to the first scenario, the dissipation mechanism is not affected and the diameter of the global attractor remains uniformly bounded with respect to the very large dispersion coefficient. However, the limit equation, as the dispersion parameter tends to infinity, becomes a gradient system. Therefore, adding the large dispersion term actually suppresses the non-trivial dynamics. According to the second scenario, neither the dissipation mechanism, nor the dynamics are essentially affected by the large dispersion and the limit dynamics remains complicated (chaotic). Finally, it is demonstrated in the third scenario that the dissipation mechanism is completely destroyed by the large dispersion, and that the diameter of the global attractor grows together with the growth of the dispersion parameter.