KLOEDEN , Peter E.

Under a one-sided dissipative Lipschitz condition on its drift, a stochastic evolution equation with additive noise of the reaction-diffusion type is shown to have a unique stochastic stationary solution which pathwise attracts all other solutions. A similar situation holds for each Galerkin approximation and each implicit Euler scheme applied...

The globally modified Navier-Stokes equations (GMNSE) were introduced by Caraballo, Kloeden & Real in 2006 and have been investigated in a number of papers since then, both for their own sake and as a means of obtaining results about the 3-dimensionalNavier-Stokes equations. These results were reviewed by Kloeden et al, which was published in...

It is shown that the synchronization of dissipative systems persists when they are disturbed by additive noise no matter how large the intensity of the noise provided asymptotically stable stationary stochastic solutions are used instead of asymptotically stable equilibria.

Negatively invariant compact sets of autonomous and nonautonomous dynamical systems on a metric space, the latter formulated in terms of processes, are shown to contain a strictly invariant set and hence entire solutions. For completeness the positively invariant case is also considered. Both discrete and continuous time systems are considered....

Under appropriate regularity conditions it is shown that the continuous dependence of the global attractors \mathcal{A}_\tau of semi dynamical systems S^{(\tau)}(t) in C([-\tau,0];Z) with Z a Banach space and time delay \tau \in [T_*,T^*], where T_* > 0, is equivalent to the equi-attraction of the attractors. Examples and counter examples posed...

En esta comunicación se presentan brevemente los resultados del trabajo [P. E. Kloeden & P. Marín-Rubio. Equi-Attraction and the continuous dependence of attractors on time delays. Discrete Contin. Dyn. Syst. Ser. A.]: condiciones de equivalencia sobre la continuidad de atractores para semiflujos generados por ecuaciones diferenciales con...

We show that infinite-dimensional integro-differential equations which involve an integral of the solution over the time interval since starting can be formulated as non-autonomous delay differential equations with an infinite delay. Moreover, when conditions guaranteeing uniqueness of solutions do not hold, they generate a non-autonomous...

Weak pullback attractors are defined for non-autonomous difference inclusions and their existence and upper semi continuous convergence under perturbation is established. Unlike strong pullback attractors, invariance and pullback attraction here are required only for (at least) a single trajectory rather than all trajectories at each starting...

Strongly negatively invariant compact sets of set-valued autonomous and nonautonomous dynamical systems on a complete metric space, the latter formulated in terms of processes, are shown to contain a weakly positively invariant family and hence entire solutions. For completeness the strongly positively invariant case is also considered, where...

The study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems. In the deterministic case, the properties of flattening and squeezing in infinite-dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply...

We investigate the existence, uniqueness and exponential stability of non-constant stationary solutions of stochastic semilinear evolution equations. Our main result shows, in particular, that noise can have a stabilization effect on deterministic equations. Moreover, we do not require any commutative condition on the noise terms.

The existence and uniqueness of global solutions for a fractional functional differential equation is established. The asymptotic behaviour of a lattice system with a fractional substantial time derivative and variable time delays is investigated. The existence of a global attracting set is established. It is shown to be a singleton set under a...

Traditional assumptions in the simple chemostat model include fixed availability of the nutrient and its supply rate, and fast flow rate to avoid wall growth. However, these assumptions become unrealistic when the availability of a nutrient depends on the nutrient consumption rate and input nutrient concentration and when the flow rate is not...

The synchronization of Stratonovich stochastic di erential equations (SDE) with a one-sided dissipative Lipschitz drift and linear multiplicative noise is investigated by transforming the SDE to random ordinary di erential equations (RODE) and synchronizing their dynamics. In terms of the original SDE this gives synchronization only when the...

The appearance of delay terms in a chemostat model can be fully justified since the future behavior of a dynamical system does not in general depend only on the present but also on its history. Sometimes only a short piece of history provides the relevant influence (bounded or finite delay), while in other cases it is the whole history that has...

We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the...

Chemostat refers to a laboratory device used for growing microorganisms in a cultured environment, and has been regarded as an idealization of nature to study competition modeling of mathematical biology. The simple form of chemostat model assumes that the availability of nutrient and its supply rate are both fixed. In addition the tendency of...

The existence and limiting behavior of the solutions of stochastic parabolic problems with thermal memory are investigate in the cases that the nonlinear term satisfies subcritical and critical growth conditions. The existence, uniqueness and continuity of solutions is proved by a semigroup method and the Lax-Milgram theorem, then the dynamics...

The existence of a pullback (and also a uniform forward) attractor is proved for a damped wave equation containing a delay forcing term which, in particular, covers the models of Sine-Gordon type. The result follows from the existence of a compact set which is uniformly attracting for the two-parameter semigroup associated to the model.

The existence and uniqueness of a variational solution satisfying energy equality is proved for a semilinear heat equation in a non-cylindrical domain with homogeneous Dirichlet boundary condition, under the assumption that the spatial domains are bounded and increase with time. In addition, the non-autonomous dynamical system generated by this...

We obtain regularity results for solutions of the three dimensional system of globally modified Navier-Stokes equations, and we investigate the relationship between global attractors, invariant measures, time-average measures and statistical solutions of these system in the case of temporally independent forcing.

The upper semi-continuous convergence of approximate attractors for an infinite delay differential equation of logistic type is proved, first for the associated truncated delay equation with finite delay and then for a numerical scheme applied to the truncated equation.

We prove the existence of a stationary random solution to a delay random ordinary differential system which attracts all other solutions in both pullback and forwards senses. The equation contains a one-sided dissipative Lipschitz term without delay, while the random delay appears in a globally Lipschitz one. The delay function only needs to be...

A new proof of existence of solutions for the three dimensional system of globally modified Navier-Stokes equations introduced in [3] by Caraballo, Kloeden and Real is obtained using a smoother Galerkin scheme. This is then used to investigate the relationship between invariant measures and statistical solutions of this system in the case of...

En esta comunicación presentaremos resultados de existencia y unicidad de soluciones que verifican una igualdad de energía para una ecuación semilineal en dominios no-cilíndricos (cf. [P. E. Kloeden, P. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations 244 (2008),...