随机反应扩散方程对系数的连续依赖性和整体解的存在性
Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations
摘要：
本文证明了半线性随机发展方程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，在巴拿赫空间B上，在希尔伯特空间H中，由柱布朗运动W_H驱动。我们假设算子A_n收敛于A，局部Lipschitz函数F_n和G_n在适当的意义下收敛于局部Lipschitz函数F和G。此外，我们还得到了极限问题解X的生存期的估计，这些估计是由近似解X_n的生存期给出的.我们应用这些结果证明了具有乘性噪声和多项式有界反应项且满足适当耗散性条件的反应扩散方程的整体解的存在性.控制方程的线性部分的算子可以是任意一致椭圆的二阶椭圆算子。
英文摘要：
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.
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