= Solution
Let $\widehat u_0$ be the <Fourier transform> of the initial <temperature>. Solving the scalar <linear ordinary differential equation> gives
$$
\boxed{\widehat u(k_x,k_y,t)=\widehat u_0(k_x,k_y)e^{-\alpha(k_x^2+k_y^2)t}.}
$$
Recover $u$ using the <Fourier inversion theorem>, with inverse factor $(2\pi)^{-2}$ for this two-dimensional convention. Equivalently, take the <convolution> of the initial data with the two-dimensional <heat kernel>. On a periodic square, the same exponential applies to each discrete <Fourier mode>, with $k_x=2\pi m/L_x$ and $k_y=2\pi n/L_y$.
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