Reference For a PDE text that treats non homogenuous boundary conditions Rigorously

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I am interested in reading a text or paper where elliptic and parabolic PDE's are discussed on bounded domains with non-homogeneous boundary conditions. I haven't been able to find anything in the usual places (Evan's and some Finite Element books). I would like to see proofs of stability/ contraction for parabolic PDE's with time varying boundary conditions.

Does anyone have a preference for me? Thanks!