 Positivity of the first Dirichlet reaction-diffusion coefficient

For nonnegative nonzero integrable initial data $f$ on $(0,L)$, the first coefficient in the normalized Fourier sine series is strictly positive, because the first sine eigenfunction is positive throughout the interval. For ${\rho }_t=D{\rho }_{xx}+\gamma \rho$ with zero Dirichlet boundary conditions, the first decay rate is ${\lambda }_1=D{\pi }^2/L^2-\gamma$. Every higher rate exceeds it by $D{\pi }^2(n^2-1)/L^2$, so smoothing for positive time and the spectral gap imply $e^{{\lambda }_{1}t}\rho (x,t)\to a_1\sqrt{2/L}\sin (\pi x/L)$ uniformly. Thus all interior points share the growth/decay threshold $\gamma =D{\pi }^2/L^2$; the endpoints remain zero. At equality the limiting profile is nonzero and stationary.