REAL ANGUAS , José

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We prove some results on the existence and uniqueness of solutions for a class of evolution equations of second order in time, containing some hereditary characteristics. Our theory is developed from a variational point of view, and in a general functional setting which permits us to deal with several kinds of delay terms. In particular, we can...

For an abstract dynamical system, we establish, under minimal assumptions, the existence of D-attractor, i.e. a pullback attractor for a given class D of families of time varying subsets of the phase space. We relate this concept with the usual attractor of fixed bounded sets, pointing out its usefulness in order to ensure the existence of this...

Sufficient conditions to get exponential stability for the sample paths (with probability one) of a non-linear monotone stochastic Partial Differential Equation are proved. In fact, we improve a stability criterion established in Chow since, under the same hypotheses, we get pathwise exponential stability instead of stability of sample paths.

We obtain a result of existence of solutions to the 2D-Navier-Stokes model with delays, when the forcing term containing the delay is sub-linear and only continuous. As a consequence of the continuity assumption the uniqueness of solutions does not hold in general. We use then the theory of multi-valued dynamical system to establish the...

The existence of an attractor for a 2D-Navier-Stokes system with delay is proved. The theory of pullback attractors is successfully applied to obtain the results since the abstract functional framework considered turns out to be nonautonomous. However, on some occasions, the attractors may attract not only in the pullback sense but in the...

Some results related to stochastic differential equations with reflecting boundary conditions (SDER) are obtained. Existence and uniqueness of strong solution is ensured under the relaxation on the drift coefficient (instead of the Lipschitz character, a monotonicity condition is supposed).

A probabilistic representation of the solution (in the viscosity sense) of a quasi-linear parabolic PDE system with non-lipschitz terms and a Neumann boundary condition is given via a fully coupled forward-backward stochastic differential equation with a reflecting term in the forward equation. The extension of previous results consists on the...

Some results on the asymptotic behaviour of solutions to Navier-Stokes equations when the external force contains some hereditary characteristics are proved. We show two different approaches to prove the convergence of solutions to the stationary one, when this is unique. The first is a direct method while the second is based in the Razumikhin type one.

We consider a stochastic non–linear Partial Differential Equation with delay which may be regarded as a perturbed equation. First, we prove the existence and the uniqueness of solutions. Next, we obtain some stability results in order to prove the following: if the unperturbed equation is exponentially stable and the stochastic perturbation is...

We prove the existence of tempered and nontempered pullback attractors for two dimensional Navier–Stokes equations on unbounded domains satisfying Poincaré inequality, for the case in which a forcing term involving memory effects appears. Our proof uses an energy method and is valid for the autonomous and nonautonomous cases.

Some results on the existence and uniqueness of solutions to Navier-Stokes equations when the external force contains some hereditary characteristics are proved.

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We prove, on one hand, that for a convenient body force with values in the distribution space (H−1(D))d, where D is the geometric domain of the fluid, there exist a velocity u and a pressure p solution of the stochastic Navier-Stokes equation in dimension 2, 3 or 4. On the other hand, we prove that, for a body force with values in the dual...

Some results on the pathwise asymptotic stability of solutions to stochastic partial differential equations are proved. Special attention is paid in proving sufficient conditions ensuring almost sure asymptotic stability with a nonexponential decay rate. The situation containing some hereditary characteristics is also treated. The results are...

Sufficient conditions for exponential mean square stability of solutions to delayed stochastic partial differential equations of second order in time are established. As a consequence of these results, some ones on the pathwise exponential stability of the system are proved. The stability results derived are applied also to partial differential...

Sufficient conditions for exponential mean square stability of solutions to delayed stochastic partial differential equations of second order in time are established. As a consequence of these results, the pathwise exponential stability of the system is also deduced. The stability results derived can be applied also to partial differential...

Some results on the existence and uniqueness of solutions for stochastic evolution equations containing some hereditary characteristics are proved. In fact, our theory is developed from a variational point of view and in a general functional setting which permit us to deal with several kinds of delay terms in a unified formulation.

Some sufficient conditions concerning stability of solutions of stochastic differential evolution equations with general decay rate are first proved. Then, these results are interpreted as suitable stabilization ones for deterministic and stochastic systems. Also, they permit us to construct appropriate linear stabilizers in some particular situations.

Stability investigation of hereditary systems often is connected with the construction of Lyapunov functionals. The general method of Lyapunov functionals construction, that was proposed by V.Kolmanovskii and L.Shaikhet and successfully used already for functional-differential equations, for difference equations with discrete time, for...