= Cooling of an initially uniform square
{title2=$T\sim(16T_0/\pi^2)\sin(\pi x/\ell)\sin(\pi y/\ell)e^{-2\kappa\pi^2t/\ell^2}$}
For a square of side $\ell$ initially at constant temperature $T_0$, held at zero boundary temperature for $t>0$, the <heat equation> $T_t=\kappa\Delta T$ with $\kappa>0$ has solution
$$
T=\frac{16T_0}{\pi^2}\sum_{\substack{m,n\geq1\\m,n\text{ odd}}}\frac{\sin(m\pi x/\ell)\sin(n\pi y/\ell)}{mn}\,e^{-\kappa\pi^2(m^2+n^2)t/\ell^2}.
$$
The <double Fourier sine series> of the initial constant gives the coefficients. The spatial modes are <Dirichlet Laplacian eigenfunctions>, so <separation of variables> gives their decay rates. At late times the $(1,1)$ mode dominates, giving the displayed profile; the next nonzero decay exponent is $10\kappa\pi^2/\ell^2$. At early times the series approaches the constant at each interior point and in <L2 space>, while the boundary remains at zero. An instantaneous change to the boundary condition therefore need not produce a solution continuous at the initial boundary in space-time.
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