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Invariant Manifolds for Random Parabolic Evolution Equations with almost sectorial operators

M. Ghani Varzaneh, F. Z. Lahbiri, S. Riedel

Abstract

In this paper, we develop a way of analyzing the random dynamics of stochastic evolution equations with a non-dense domain. Such problems cover several types of evolution equations. We are particularly interested in evolution equations with non-homogeneous boundary conditions of white noise type. We prove the existence of stable, unstable, and center manifolds around a stationary trajectory by combining integrated semigroup theory and invariant manifold theory. The results are applied to stochastic parabolic equations with white noise at the boundary.

Invariant Manifolds for Random Parabolic Evolution Equations with almost sectorial operators

Abstract

In this paper, we develop a way of analyzing the random dynamics of stochastic evolution equations with a non-dense domain. Such problems cover several types of evolution equations. We are particularly interested in evolution equations with non-homogeneous boundary conditions of white noise type. We prove the existence of stable, unstable, and center manifolds around a stationary trajectory by combining integrated semigroup theory and invariant manifold theory. The results are applied to stochastic parabolic equations with white noise at the boundary.

Paper Structure

This paper contains 7 sections, 21 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of the Random Dynamical System and Well-Posedness
  4. Differentiability, compactness, and some a priori bounds
  5. Invariant Manifolds
  6. Existence of stationary points.
  7. Examples: Heat Equation with white noise at the boundary

Key Result

Lemma 2.1

Let ${A}:D({A})\subset{X}\to{X}$ be a linear operator on $({X},\|\cdot\|)$. Assume that there exist two constants, $\omega\in\mathbb{R}$ and $M>0$, such that $(\omega,+\infty)\subset\rho({A})$ and for each $\lambda>\omega$ Then the following properties are equivalent:

Theorems & Definitions (48)

  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 38 more