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

Abstract

We prove convergence of the solutions Xn of semilinear stochastic evolution equations on a Banach space B, driven by a cylindrical Brownian motion in a Hilbert space H,dXn(t)=(AnX(t)+Fn(t,Xn(t)))dt+Gn(t,Xn(t))dWH(t),Xn(0)=ξn, assuming that the operators An converge to A and the locally Lipschitz functions Fn and Gn 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 Xn.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.