Maximum entropy principle and classical evolution equations with source terms

We devise a maximum entropy technique to construct (approximate) time-dependent solutions to evolution equations endowed with source terms and, consequently, not preserving normalization. In some special cases the method yields exact solutions. It is shown that the present implementation of the maximum entropy prescription always (even in the case of approximate solutions) preserves the exact functional relationship between the time derivative of the entropy and the time-dependent solutions of the evolution equation. Other properties of the maximum entropy solutions and some illustrative examples are also discussed.