Regularizing effect of the heat equation
Solution 1:
I think the answer is no, since the analogous result doesn't even hold for elliptic equations (consider the discussion here Counterexample for the solvability of $-\Delta u = f$ for $f\in C^2$ ).
I think the answer is no, since the analogous result doesn't even hold for elliptic equations (consider the discussion here Counterexample for the solvability of $-\Delta u = f$ for $f\in C^2$ ).