  We study unique solvability for one dimensional stochastic pressure equation
with diffusion coefficient given by the Wick exponential of log-correlated
Gaussian fields. We prove well-posedness for Dirichlet, Neumann and periodic
boundary data, and the initial value problem, covering the cases of both the
Wick renormalization of the diffusion and of point-wise multiplication. We
provide explicit representations for the solutions in both cases, characterized
by the $S$-transform and the Gaussian multiplicative chaos measure.
