= Positivity of the first Dirichlet reaction-diffusion coefficient
{title2=$a_1=\sqrt{2/L}\int_0^Lf(x)\sin(\pi x/L)\,dx>0$}
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 $e^{\lambda_1t}\rho(x,t)\to a_1\sqrt{2/L}\sin(\pi x/L)$ 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.
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