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Heat Equation on $[0,T] \times \mathbb{R}^n$
See Linear and quasilinear equations of parabolic type, 1968, by O.A. Ladyzenskaja, V.A. Solonnikov, and N.N. Ural'ceva, Theorem 6.1 in Chapter III.
See also Theorem 9.1, Chapter IV, in the same book, …
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Accepted
Schauder regularity heat equation
For a partial answer, see Theorem 6.48 in Second order parabolic differential equations, 1996, by Gary Lieberman.
There is an additional assumption that $f$ belongs to the Morrey space $M^{1,n+1+\alp …