In a process to write a review I want your help in figuring out the list of possible methods by which we may prove existence and may be uniqueness of a partial differential equation, particularly the parabolic type equation. Please also mention in your answer whether the method is applicable for non-linear PDEs or not. Moreover, if possible also mention, for which types of domains (bounded/unbounded/C^1/C^\infinity/Lipschitz) does the method works. By the way, just to clarify, we are talking about analytical solutions only. Thanks.
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