The paper is concerned with a class of parabolic equations with a gradient-dependent nonlinear term in a Gauss-Sobolev space setting. Un-der a local Lipschitz continuity condition, it is shown in Theorem 4.2 that there exists a unique strong solution of such a semilinear parabolic equation for which a certain energy inequality holds. The theorem is applied to show the existence of the strong solutions to the Kolmogorov equation and the Hamilton-Jacobi-Bellman equation arising from the control problem for stochastic partial differential equations
