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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?

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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

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    $\begingroup$ Yes, I am looking at linear theory. Thank you for the reference. $\endgroup$
    – lost1
    Commented Feb 12, 2014 at 22:38

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