Antonio M. Márquez-Durán

On the Convergence of Solutions of Globally Modified Navier-Stokes Equations with Delays to Solutions of Navier-Stokes Equations with Delays

Abstract We prove that under suitable assumptions, from a sequence of solutions of Globally Modified Navier-Stokes equations with delays we can extract a subsequence which converges in an adequate sense to a weak solution of a three-dimensional Navier-Stokes equation with delays. An additional case with a family of different delays involved in the approximating problems is also discussed.

Well-Posedness and Asymptotic Behavior of a Nonclassical Nonautonomous Diffusion Equation with Delay

In this paper, a nonclassical nonautonomous diffusion equation with delay is analyzed. First, the well-posedness and the existence of a local solution is proved by using a fixed point theorem. Then, the existence of solutions defined globally in future is ensured. The asymptotic behavior of solutions is analyzed within the framework of pullback attractors as it has revealed a powerful theory to describe the dynamics of nonautonomous dynamical systems. One difficulty in the case of delays concerns the phase space that one needs to construct the evolution process. This yields the necessity of using a version of the Ascoli–Arzela theorem to prove the compactness.

Some Results on Nonlinear Backward Stochastic Evolution Equations

Abstract In this paper, we present some results concerning existence and uniqueness of solutions for a rather general class of nonlinear backward stochastic partial differential equations. These results are illustrated with two examples.

A Nonclassical and Nonautonomous Diffusion Equation Containing Infinite Delays

We first study the well-posedness of a nonclassical and nonautonomous diffusion equation containing unbounded delays. Then, we prove the existence and uniqueness of local solutions, and finally we prove the global in time existence of solutions as well as the continuous dependence on the initial values.

Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic

Our aim in this work is the study of the existence and uniqueness of solutions for a non-classical and non-autonomous diffusion equation containing infinite delay terms. We also analyze the asymptotic behaviour of the system in the pullback sense and, under suitable additional conditions, we obtain global exponential decay of the solutions of the evolutionary problem to stationary solutions.