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On mild solutions to some dissipative SPDEs on $L^p$ spaces with additive noise

Carlo Marinelli

Abstract

We establish well-posedness in the mild sense for a class of stochastic semilinear evolution equations on $L^p$ spaces on bounded domains of $\mathbb{R}^n$ with a nonlinear drift term given by the superposition operator generated by a monotone function on the real line with power-like growth. The noise is of additive type with respect to a cylindrical Wiener process, with diffusion coefficient not necessarily of $γ$-Radonifying type.

On mild solutions to some dissipative SPDEs on $L^p$ spaces with additive noise

Abstract

We establish well-posedness in the mild sense for a class of stochastic semilinear evolution equations on spaces on bounded domains of with a nonlinear drift term given by the superposition operator generated by a monotone function on the real line with power-like growth. The noise is of additive type with respect to a cylindrical Wiener process, with diffusion coefficient not necessarily of -Radonifying type.
Paper Structure (13 sections, 21 theorems, 142 equations)

This paper contains 13 sections, 21 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Some elementary inequalities
  5. Duality mappings
  6. Estimates for mild solutions
  7. A null sequence
  8. Existence and uniqueness of solutions
  9. Estimates of solutions to the regularized equation
  10. Convergence of approximating solutions
  11. Existence and uniqueness
  12. Further properties of mild solutions
  13. Generalized solutions

Key Result

Lemma 2.1

Let $x,y \in \mathbb{R}_+$. If $a \in [0,1]$, then If $a \in [1,\infty\mathclose[$, then

Theorems & Definitions (45)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 35 more