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**Da Prato-Zabczyk’s maximal inequality revisited. I.**
*(English)*
Zbl 0785.35115

The author studies the problem of existence, uniqueness and regularity of mild solutions to semilinear non-autonomous stochastic parabolic equations. Here locally Lipschitzian nonlinear terms are considered. One of the significant points of this work stems from the use of factorization method in the proofs.

Similar results in the linear case are extended in three ways: 1) in non- autonomous problems, the semigroups in the definition of a mild solution are replaced by two-parameter evolution systems; 2) in the parabolic case with state-dependent diffusion coefficient, existence and regularity of solutions to equations whose nonlinear terms are defined and Lipschitzian on a Banach space embedded continuously into a Hilbert space are established; and 3) existence of solutions to equations the coefficients of which are Lipschitz only on bounded sets in a Banach space is also proved.

Similar results in the linear case are extended in three ways: 1) in non- autonomous problems, the semigroups in the definition of a mild solution are replaced by two-parameter evolution systems; 2) in the parabolic case with state-dependent diffusion coefficient, existence and regularity of solutions to equations whose nonlinear terms are defined and Lipschitzian on a Banach space embedded continuously into a Hilbert space are established; and 3) existence of solutions to equations the coefficients of which are Lipschitz only on bounded sets in a Banach space is also proved.

Reviewer: K.M.Ramachandran (Tampa)

### MSC:

35R60 | PDEs with randomness, stochastic partial differential equations |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |