Strong solutions of stochastic equations with singular time dependent drift

We prove existence and uniqueness of strong solutions to stochastic equations in domains G subset of R-d with unit diffusion and singular time dependent drift b up to an explosion time. We only assume local L-q-L-p-integrability of b in R x G with d/ p+ 2/ q < 1. We also prove strong Feller properties in this case. If b is the gradient in x of a nonnegative function psi blowing up as G there exists x --> partial derivativeG, we prove that the conditions 2D(t) psi less than or equal to Kpsi, 2D(t)psi + Deltapsi less than or equal to Ke(epsilonpsi), epsilon is an element of [0, 2), imply that the explosion time is infinite and the distributions of the solution have sub Gaussian tails.