In the answer to <https://math.stackexchange.com/questions/166286/viscosity-solution-vs-weak-solution> 


H. Ishii, "On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions", Funkcial Ekvac. Ser. Int. 38 (1) (1995) 101–120.(pdf)


is mentioned. This holds in the elliptic case. Do we know anything about parabolic equations?



Equivalence of distributional and viscosity solution in parabolic case?
In general this probably depends on the parabolic equation you are studying. I think for some nonlinear equations the equivalence is known not to be true as the equations may have unbounded weak solutions. 


If you are, however, looking for the linear theory such results probably exist although I do not know what is the most general structure you can have. In any case, I would start by studying the below paper and the references therein.


<http://users.jyu.fi/~peanju/preprints/equivalence.pdf>


