Non-uniqueness of solutions of the heat equation
Solution 1:
The type of solutions you are looking for were constructed by Tychonov. You can find them in chapter 7 of Fritz John's classical book Partial Differential Equations.
Uniqueness follows also from one sided inequalities like in Widder's theorem: if $f(x,t)$ is a non-negative solution of the heat equation with $f(x,0)=0$, then $f\equiv0$. You can find a proof again in chapter 7 of Fritz John's book.