For the heat equation $(\partial\_t-\partial\_x^2)f(t,x)=0$ defined on $[0,T)\times(-\infty,\infty)$, to obtain uniqueness of the initial value problem, usually it is required to limit the growth of the potential solution at infinity, i.e. $|f(t,x)|<\exp(c\cdot x^2)$. 
My question is, if we do not impose any such conditions, is uniqueness no longer valid? In particular, is there a well known example of a function $f(t,x)$ that satisfies the heat equation on $[0,T)\times(-\infty,\infty)$, $f(0,x)=0$, but $f$ is not identically zero? 


What if we relaxed the conditions a little bit, and only required that $f$ satisfies the heat equation in $(0,T)\times(-\infty,\infty)$ and is continuous on $[0,T)\times(-\infty,\infty)$, is there an example in this case?



Non-uniqueness of solutions of the heat equation
Tychonoff in his 1935 paper [Théorèmes d'unicité pour l'équation de la chaleur](http://mi.mathnet.ru/eng/msb/v42/i2/p199) proved uniqueness if the solutions are not too large, and gave an example to show that the solution is not unique in general. His counterexample grows extremely rapidly for large x. 


