Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations
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.
Cite
@article{arxiv.1104.4258,
title = {Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations},
author = {Markus Kunze and Jan van Neerven},
journal= {arXiv preprint arXiv:1104.4258},
year = {2019}
}
Comments
A confusing typo in the statement of Theorem 4.9 has been corrected