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It is well known that there is a heat kernel (or fundamental solution) of the Cauchy problem for the heat equation on $\mathbb{R}^{n}$. I have a simple question. How do I show that the fundamental solution $f(t,x)$ satisfies $\lim_{|x|\rightarrow \infty}f(t,x) = 0$ for any fixed t $\gt$ 0

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Something is wrong with your question: you cannot fix $x$ if it is a dummy variable. Do you mean "for any fixed $t > 0$"? Or perhaps the limit is for $t\to 0$? – Giuseppe Negro Nov 2 '12 at 21:21
It should indeed be 'for any fixed t>0' – Peadar Coyle Nov 7 '12 at 0:06
up vote 1 down vote accepted

You have to show that for any small $\epsilon>0$ that you can find a $t$, or an $x$ depending on what you are asking, and show that the kernel is smaller than $\epsilon$ for any time or point larger than $t$, or $x$ when the other variable is fixed. The idea is that the error in approximating the kernel with $0$ can be made as small as you like by looking at further points or larger times.

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