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: { d X ( t ) = [ A ( t , X ( t ) ) + f ( t , X ( t ) ) ] d t + g ( t , X ( t ) ) d W ( t ) , t ∈ [ 0 , T ] , X ( 0 ) = X 0 ∈ H , where A ( t , ⋅ ) : V → V * is a linear bounded operator with coercivity, monotone condition and hemicontinuity, f : [ 0 , ∞ ) × H → H and g : [ 0 , ∞ ) × 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].