The Heat Equation and the Fourier transform: a paradox
Paradox (noun) a statement or proposition that seems
self-contradictory or absurd, but in reality expresses a possible
truth.
Solution by Fourier transform
To solve the heat equation
\[
u_t = \Delta u, \qquad u(x, 0) = g(x)
\]
we consider the Fourier transform
\[
\hat u(\xi, t) = (2\pi)^{-n/2} \int e^{-ix\cdot \xi} u(x, t) \, dx.
\]
It satisfies
\[
\hat u_t = -|\xi|^2 \hat u(\xi, t), \qquad \hat u(\xi, 0) = \hat g(\xi),
\]
so that $\hat u(\xi, t) = e^{-t|\xi|^2} \hat g(\xi)$. The solution
of the heat equation can be recovered by applying the inverse Fourier transform:
\[
u(x, t) = (2\pi)^{-n/2} \int e^{ix\cdot \xi} e^{-t|\xi|^2} \hat g(\xi)\, d\xi.
\]
Since $e^{-t|\xi|^2}$ is a smooth function which vanishes very
rapidly as $|\xi|\to\infty$, multiplication with $e^{-t|\xi|^2}$
only improves the function $ e^{-t|\xi|^2} \hat g(\xi)$, so the
above solution is defined for all $t\ge0$.
Conclusion: solutions to the heat equation never become
singular for $t\gt0$.
A singular solution
The function
\[
u(x, t) = \frac{1} {\bigl(4\pi(1-t)\bigr)^{n/2}} e^{|x|^2/4(1-t)}
\]
is a smooth solution to the heat equation which becomes singular as
$t\nearrow1$.
Conclusion: solutions to the heat equation can become
singular at some $t\gt0$.