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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 solutionThe 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.

Multiply by the $(m,n)$ sine product and integrate over the square. Fourier orthogonality gives squared norm ${\ell }^2/4$, and thusThereforeThis double Fourier sine series equals one at interior points. At boundary points every summand vanishes, consistent with the endpoint value of the sine extension.