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
Analysis of the fundamental solution to the heat equation
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0It should indeed be 'for any fixed t>0' – 2012-11-07
1 Answers
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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.