A one-dimensional evolution equation transformable into a linear one coupled to a quadratic Smoluchowski (an Ornstein-Uhlenbeck) noise is considered. A one-dimensional probability distribution is obtained by way of a characteristic function which is expressed by functionals of the Smoluchowski process. It is shown that in the frame of the presented approach the probability density can be found only for a particular value of the damping constant in the linear-type relaxation equation, It is also shown that in a special case the white noise limit may be performed.