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Pathwise mild solutions for superlinear stochastic evolution equations and their attractors

Alexandra Blessing, Tim Seitz, Stefanie Sonner, Bao Quoc Tang

Abstract

We investigate stochastic parabolic evolution equations with time-dependent random generators and locally Lipschitz continuous drift terms. Using pathwise mild solutions, we construct an infinite-dimensional stationary Ornstein-Uhlenbeck type process, which is shown to be tempered in suitable function spaces. This property, together with a bootstrapping argument based on the regularizing effect of parabolic evolution families, is then applied to prove the global well-posedness and the existence of a random attractor for reaction-diffusion equations with random non-autonomous generators and nonlinearities satisfying certain growth and dissipativity assumptions.

Pathwise mild solutions for superlinear stochastic evolution equations and their attractors

Abstract

We investigate stochastic parabolic evolution equations with time-dependent random generators and locally Lipschitz continuous drift terms. Using pathwise mild solutions, we construct an infinite-dimensional stationary Ornstein-Uhlenbeck type process, which is shown to be tempered in suitable function spaces. This property, together with a bootstrapping argument based on the regularizing effect of parabolic evolution families, is then applied to prove the global well-posedness and the existence of a random attractor for reaction-diffusion equations with random non-autonomous generators and nonlinearities satisfying certain growth and dissipativity assumptions.

Paper Structure

This paper contains 11 sections, 22 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Random dynamical systems
  4. Pathwise mild solutions
  5. Abstract setting and an Ornstein-Uhlenbeck-type process
  6. Local existence of solutions
  7. Temperedness of an Ornstein-Uhlenbeck-type process
  8. Random attractors for reaction-diffusion equations
  9. Global existence
  10. Random attractors
  11. Generalization for higher-order elliptic operators

Key Result

Lemma 2.2

(Gess2011) There exists a $(\theta_t)_{t\in \mathbb{R}}$-invariant subset $\Omega\subset C_0(\mathbb{R};\mathcal{X})$ of full measure, with the following properties:

Theorems & Definitions (53)

  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 43 more