Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations
Copyright policy )Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations
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.
A confusing typo in the statement of Theorem 4.9 has been corrected
- Delft University of Technology Netherlands
- TECHNISCHE UNIVERSITEIT DELFT Netherlands
global existence, Stochastic reaction diffusion equations, Continuous dependence on the coefficients, stochastic reaction diffusion equations, Probability (math.PR), stochastic evolution equations, Theoretical approximation in context of PDEs, Functional Analysis (math.FA), Mathematics - Functional Analysis, 60H15 (Primary) 35K37, 35A35, 35R60 (Secondary), Stochastic partial differential equations (aspects of stochastic analysis), Reaction-diffusion equations, FOS: Mathematics, continuous dependence on the coefficients, PDEs with randomness, stochastic partial differential equations, Global existence, Stochastic evolution equations, Analysis, Mathematics - Probability
global existence, Stochastic reaction diffusion equations, Continuous dependence on the coefficients, stochastic reaction diffusion equations, Probability (math.PR), stochastic evolution equations, Theoretical approximation in context of PDEs, Functional Analysis (math.FA), Mathematics - Functional Analysis, 60H15 (Primary) 35K37, 35A35, 35R60 (Secondary), Stochastic partial differential equations (aspects of stochastic analysis), Reaction-diffusion equations, FOS: Mathematics, continuous dependence on the coefficients, PDEs with randomness, stochastic partial differential equations, Global existence, Stochastic evolution equations, Analysis, Mathematics - Probability
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