# Non-uniqueness of solutions of the heat equation

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 My question is, if we do not impose any such conditions, is uniqueness no longer valid?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?

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There's loads of counterexamples. See my answer to the following question, and the comments. mathoverflow.net/questions/72195. Also, Terry Tao links to the paper by Tychonoff in the question (same as Richard Borcherds' link below) – George Lowther Dec 1 '11 at 21:20
I see that this question was also asked on math.stackexchange (math.stackexchange.com/q/87464). Posting the same question on both sites at the same time is not generally considered to be a good thing to do. – George Lowther Dec 2 '11 at 1:04
Sorry, I'll avoid doing that in the future. In my defense, I wasn't sure which site was more appropriate for this question. – Ivan Dec 2 '11 at 5:47