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

Let H,VH,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∈[0,T],X(0)=X0∈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→L20(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].