= Solution
For $T_n=A\cos(k_nx)e^{-\kappa k_n^2t}$, direct <differentiation> again gives $\partial_tT_n=-\kappa k_n^2T_n$ and $\partial_x^2T_n=-k_n^2T_n$. Thus it satisfies the <heat equation>. Moreover,
$$
\partial_xT_n=-Ak_n\sin(k_nx)e^{-\kappa k_n^2t}
$$
vanishes at both endpoints for integer $n$, proving the homogeneous <Neumann boundary conditions>. Now $n=0$ gives the nonzero constant mode $T=A$, which does not decay. It represents conserved mean <temperature> in an insulated plate.
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