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Local solution to an energy critical 2-D stochastic wave equation with exponential nonlinearity in a bounded domain

Zdzisław Brzeźniak, Nimit Rana

Abstract

We prove the existence and the uniqueness of a local maximal solution to an $H^1$-critical stochastic wave equation with multiplicative noise on a smooth bounded domain $\mathcal{D} \subset \mathbb{R}^2$ with exponential nonlinearity. First, we derive the appropriate deterministic and stochastic Strichartz inequalities in suitable spaces and, then use them in arguments based on fixed point method to show the local well-posedness result. We also present an explosion result for the constructed unique local maximal solution.

Local solution to an energy critical 2-D stochastic wave equation with exponential nonlinearity in a bounded domain

Abstract

We prove the existence and the uniqueness of a local maximal solution to an -critical stochastic wave equation with multiplicative noise on a smooth bounded domain with exponential nonlinearity. First, we derive the appropriate deterministic and stochastic Strichartz inequalities in suitable spaces and, then use them in arguments based on fixed point method to show the local well-posedness result. We also present an explosion result for the constructed unique local maximal solution.

Paper Structure

This paper contains 24 sections, 41 theorems, 323 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Function spaces and interpolation theory
  4. Stochastic analysis
  5. Various types of measurability
  6. Stopping times
  7. Inhomogeneous Strichartz estimates
  8. Stochastic Strichartz estimates
  9. Main assumptions
  10. Martingales
  11. Burkholder inequality
  12. Local well-posedness - preliminary results
  13. SNLWE and assumptions
  14. Definition of a local mild solution
  15. The approximation problem
  16. ...and 9 more sections

Key Result

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With our notation from this section, we have the following

Theorems & Definitions (103)

  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
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  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 93 more