A Note on non-autonomous scalar functional differential equations with small delay

We prove that the minimal sets in the skew-product semiflows generated from a non-autonomous scalar functional differential equation with a small delay are all almost automorphic extensions of the base. This result is not true for arbitrary delay equations. The point is that, for a small delay, so-called special solutions exist and permit us to tackle the problem by means of some related scalar ODE's for which the study is much simpler. To cite this article: A.I. Alonso et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Well-posedness of non-autonomous linear evolution equations for generators whose commutators are scalar

We prove the well-posedness of non-autonomous linear evolution equations for generators \({A(t): D(A(t)) \subset X \to X}\) whose pairwise commutators are complex scalars, and in addition, we establish an explicit representation formula for the evolution. We also prove well-posedness in the more general case where instead of the onefold commutators only the p-fold commutators of the operators A(t) are complex scalars. All these results are furnished with rather mild stability and regularity assumptions: Indeed, stability in X and strong continuity conditions are sufficient. Additionally, we improve a well-posedness result of Kato for group generators A(t) by showing that the original norm continuity condition can be relaxed to strong continuity. Applications include Segal field operators and Schrödinger operators for particles in external electric fields.     

Periodic solutions in general scalar non-autonomous models with delays

Theorems for the existence of periodic solutions for diverse models of population dynamics are obtained as corollaries of a few basic theorems, thus unifying the analysis of a broad class of scalar models in a single setting. The latter mechanism allows to obtain existence conditions for a broad class of nonlinear, non-autonomous models and models with state-dependent delays. The technique fulfills multiple roles: it can be used to expand on well-known results as well as to shorten existing proofs. We provide some examples which illustrate the applicability of our results.     

Operator splitting for non-autonomous evolution equations

We establish general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tools are evolution semigroups allowing the direct application of existing results on autonomous problems. The results obtained are illustrated by the example of an autonomous diffusion equation perturbed with time dependent potential. We also prove convergence rates for the sequential splitting applied to this problem.     

Bifurcations in reaction-diffusion equations and associated dissipative structures

The paper considers semilinear parabolic equations with conditions dependent on a parameter. A stationary solution of the boundary-value problem is constructed in the neighborhood of the bifurcation value of the parameter. The evolution of the solutions of the Cauchy problem to the bifurcation solution—a spatially nonhomogeneous dissipative structure—is examined. Translated from Chislennye Metody i Vychislitel'nyi Eksperiment, Moscow State University, pp. 15–30, 1998.     

A trichotomy result for non-autonomous rational difference equations

We study non-autonomous rational difference equations. Under the assumption of a periodic non-autonomous parameter, we show that a well known trichotomy result in the autonomous case is preserved in a certain sense which is made precise in the body of the text. In addition we discuss some questions regarding whether periodicity preserves or destroys boundedness.     

Solutions to non-autonomous integrodifferential equations with infinite delay

The conclusions of Da Prato and Sinestrari concerning the non-autonomous evolution operator of hyperbolic type for the equation

     

High-order splitting methods for separable non-autonomous parabolic equations

We consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative coefficients). We propose to consider a class of methods that allows us to evaluate all time-dependent operators at real values of the time, leading to schemes which are stable and simple to implement. If the system can be considered as the perturbation of an exactly solvable problem and the flow of the dominant part is advanced using real coefficients, it is possible to build highly efficient methods for these problems. We show the performance of this class of methods on several numerical examples and present some new improved schemes.     

Uniform attractors for non-autonomous Brinkman-Forchheimer equations with delay

In this paper, we prove the existence of a uniform attractor for non-autonomous Brinkman-Forchheimer equations with general delay and time-dependent external force.     

Kernel sections for non-autonomous strongly damped wave equations

We prove the existence of compact kernel sections for the process generated by a non-autonomous strongly damped wave equation with homogeneous Dirichlet boundary condition. We show that the upper bound of the Hausdorff dimension of sections decreases as the damping grows for large strong damping.     

GLOBAL ATTRACTIVITY OF LINEAR NON－AUTONOMOUS NEUTRAL DIFFERENTIAL－DIFFERENCE EQUATIONS

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