The existence and uniqueness of energy solutions to local non-Lipschitz stochastic evolution equations

Let H, V and K be separable Hilbert spaces. In this paper we consider the existence and uniqueness of energy solutions to the following stochastic evolution equation: {dX(t)=[A(t, X(t)) + f(t, X(t))] dt + g(t, X(t))dW(t), t is an element of [0, T], X(0) = X(0) is an element of H, where A(t,.): V -> V* is a linear bounded operator with coercivity, monotone condition and hemicontinuity, f : [0, infinity) x H -> H and g: [0, infinity) x H -> L(2)(0)(K, H) are measurable functions and satisfy the local non-Lipschitz condition proposed by the author [T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differential Equations 96 (1992) 152-169]. (C) 2009 Elsevier Inc. All rights reserved.