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

## TL;DR

This paper establishes the local existence and uniqueness of solutions for a 2-D stochastic wave equation with exponential nonlinearity, using Strichartz inequalities and fixed point methods, and discusses potential blow-up scenarios.

## Contribution

It provides the first local well-posedness result for an energy-critical stochastic wave equation with exponential nonlinearity in two dimensions.

## Key findings

- Proved local existence and uniqueness of solutions.
- Derived Strichartz inequalities for stochastic wave equations.
- Identified conditions leading to solution blow-up.

## 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.