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 dXn(t) = (AnX(t) + Fn(t,Xn(t)))dt + Gn(t,Xn(t))dWH(t), Xn(0) = xn, on a Banach space B, driven by a cylindrical Brownian motion WH in a Hilbert space H. We assume 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.