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Nonlinear semigroups for delay equations in Hilbert spaces, inertial manifolds and dimension estimates

Mikhail Anikushin

Abstract

We study the well-posedness of nonautonomous nonlinear delay equations in $\mathbb{R}^{n}$ as evolutionary equations in a proper Hilbert space. We present a construction of solving operators (nonautonomous case) or nonlinear semigroups (autonomous case) for a large class of such equations. The main idea can be easily extended for certain PDEs with delay. Our approach has lesser limitations and much more elementary than some previously known constructions of such semigroups and solving operators based on the theory of accretive operators. In the autonomous case we also study differentiability properties of these semigroups in order to apply various dimension estimates using the Hilbert space geometry. However, obtaining effective dimension estimates for delay equations is a nontrivial problem and we explain it by means of a scalar delay equation. We also discuss our adjacent results concerned with inertial manifolds and their construction for delay equations.

Nonlinear semigroups for delay equations in Hilbert spaces, inertial manifolds and dimension estimates

Abstract

We study the well-posedness of nonautonomous nonlinear delay equations in as evolutionary equations in a proper Hilbert space. We present a construction of solving operators (nonautonomous case) or nonlinear semigroups (autonomous case) for a large class of such equations. The main idea can be easily extended for certain PDEs with delay. Our approach has lesser limitations and much more elementary than some previously known constructions of such semigroups and solving operators based on the theory of accretive operators. In the autonomous case we also study differentiability properties of these semigroups in order to apply various dimension estimates using the Hilbert space geometry. However, obtaining effective dimension estimates for delay equations is a nontrivial problem and we explain it by means of a scalar delay equation. We also discuss our adjacent results concerned with inertial manifolds and their construction for delay equations.

Paper Structure

This paper contains 7 sections, 31 theorems, 100 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Construction of delay semigroups
  3. Differentiability of delay semigroups
  4. Example
  5. Inertial manifolds
  6. Example (continued)
  7. An addition to the reduction principle

Key Result

Theorem 1.1

Suppose for EQ: ClassicalDelayEquation that $F$ satisfies EQ: DelayLipschitz. Then for any $t_{0} \in \mathbb{R}$ and $v_{0} \in \mathbb{H}$ there is a unique generalized solution $v(t)=v(t,t_{0},v_{0})$ to EQ: AbstractDelayHilberSpace, which is a continuous function $[t_{0},+\infty) \to \mathbb{H}$

Figures (2)

  • Figure 1: A numerically obtained region in the space of parameters $(\tau,\alpha)$ of system \ref{['EQ: ElNinoSSmodel']}, where \ref{['EQ: SSmodelRoughMuEstimate']} is satisfied (orange) and \ref{['EQ: SSLambda1PlusLambda2Ineq']} is satisfied (orange and blue).
  • Figure 2: A numerically obtained region in the space of parameters $(\tau,\alpha)$ of system \ref{['EQ: ElNinoSSmodel']} for which the conditions of Theorem \ref{['TH: SSmodelOneDimIMSharper']} (orange) or Theorem \ref{['TH: SSmodelTwoDimIMSharper']} (blue) are satisfied.

Theorems & Definitions (62)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Lemma 2.1
  • Remark 3
  • Remark 4
  • Lemma 2.2
  • Lemma 2.3
  • Corollary 1
  • Remark 5
  • ...and 52 more