Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations

TL;DR

This paper establishes the continuous dependence of solutions on coefficients and proves global existence for stochastic reaction diffusion equations with multiplicative noise, under certain dissipativity and convergence conditions.

Contribution

It introduces a framework for analyzing the convergence of solutions to stochastic evolution equations with varying coefficients and applies it to reaction diffusion equations with noise.

Findings

Solutions depend continuously on coefficients and initial data.

Global existence is proven for a broad class of reaction diffusion equations.

Provides estimates for the lifetime of solutions based on approximations.

Abstract

We prove convergence of the solutions X_n of semilinear stochastic evolution equations dX_n(t) = (A_nX(t) + F_n(t,X_n(t)))dt + G_n(t,X_n(t))dW_H(t), X_n(0) = x_n, on a Banach space B, driven by a cylindrical Brownian motion W_H in a Hilbert space H. We assume that the operators A_n converge to A and the locally Lipschitz functions F_n and G_n converge to the locally Lipschitz functions F and G in an appropriate sense. Moreover, we obtain estimates for the lifetime of the solution X of the limiting problem in terms of the lifetimes of the approximating solutions X_n. We apply the results to prove global existence for reaction diffusion equations with multiplicative noise and a polynomially bounded reaction term satisfying suitable dissipativity conditions. The operator governing the linear part of the equation can be an arbitrary uniformly elliptic second order elliptic operator.