= Solution
For admissible initial data, for example a <square-integrable function> on the interval, expand in the sine <eigenfunctions> of the Dirichlet <Laplacian>:
$$
u(x,0)=\sum_{j\ge1}c_j\sin(j\pi x),\qquad
u(x,t)=\sum_{j\ge1}c_j e^{(\kappa-j^2\pi^2)t}\sin(j\pi x).
$$
The expansion follows either from <separation of variables> or the complete sine <orthogonal basis>. If $\kappa<\pi^2$, every growth rate is negative and <Parseval's identity> gives
$$
\|u(\cdot,t)\|_{L^2}^2=\frac12\sum_{j\ge1}|c_j|^2e^{2(\kappa-j^2\pi^2)t}
\le e^{-2(\pi^2-\kappa)t}\|u(\cdot,0)\|_{L^2}^2.
$$
Thus all finite-energy data decay. The convergence is also uniform in $x$ after any fixed positive time: apply <Cauchy-Schwarz inequality> to the sine series, using the summable factors $e^{-2(j^2-1)\pi^2t}$, to bound its supremum by a constant times $e^{-(\pi^2-\kappa)t}$ for $t\ge t_0>0$.
Conversely choose $u(x,0)=\sin\pi x$. Then $u(x,t)=e^{(\kappa-\pi^2)t}\sin\pi x$, which is stationary at equality and grows when $\kappa>\pi^2$. \b[All initial conditions decay if and only if $\kappa<\pi^2$.] This is the continuous part of the <reaction-diffusion spectral decay threshold>.
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