the fundamental solution of heat equations

Question

Let $$\Phi(x,t)=\frac{1}{\sqrt{4\pi t}}e^{-x^2/4t}$$ be the fundamental solution of the heat equation (or the heat kernel).

What is the supremum of $\Phi$ over $x$: $$\sup_x \Phi(x,t)?$$

Answer 1

Note that, we have $\exp(\xi) < 1$ for $\xi < 0$ and $\exp(0) = 1$. Hence, for every $x \ne 0$, we have $$ \Phi(x,t) = \frac 1{(4\pi t)^{1/2}}\exp\left(-\frac{x^2}{4t}\right) < \frac 1{(4\pi t)^{1/2}} $$ As $$ \Phi(0, t) = \frac 1{(4\pi t)^{1/2}}$$ we have that $$ \sup_{x\in \mathbf R} \Phi(x,t) = \frac 1{(4\pi t)^{1/2}} $$