 Solution

For admissible initial data, for example a square-integrable function on the interval, expand in the sine eigenfunctions of the Dirichlet Laplacian: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 givesThus 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 $e^{-2(j^2-1){\pi }^{2}t}$, 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 $\kappa >{\pi }^2$. All initial conditions decay if and only if $\kappa <{\pi }^2$. This is the continuous part of the reaction-diffusion spectral decay threshold.