Some criteria for the mean square and almost sure exponential stability of the zero solution of the nonlinear stochastic partial differential equation
are given, where , , is a Banach space and are real separable Banach spaces such that , where the injections are continuous and dense. The coefficients are assumed to satisfy the usual coercivity, boundedness, monotonicity, hemicontinuity and measurability conditions. The main results obtained by T. Caraballo and J. Real [Stochastic Anal. Appl. 12, No. 5, 517-525 (1994; Zbl 0808.93069)] are improved, which in particular concerns the case of non-autonomous equation. Several illustrating examples are also given.