= Solution
Let $f(x)=\rho(x,0)$. The first normalized <Fourier sine series> coefficient is
$$
a_1=\sqrt{\frac2L}\int_0^Lf(x)\sin\frac{\pi x}{L}\,dx>0.
$$
Strict positivity follows because $f\geq0$, has positive total mass, and $\sin(\pi x/L)>0$ everywhere inside the interval. In particular, there is no cancellation that could remove the slowest term. This is the <positivity of the first Dirichlet reaction-diffusion coefficient>.
For finite initial mass $M=\int_0^Lf(x)\,dx$, all coefficients satisfy $|a_n|\leq\sqrt{2/L}\,M$. The <eigenfunction expansion> therefore gives the uniform bound
$$
\left|e^{\lambda_1t}\rho(x,t)-a_1\psi_1(x)\right|\leq\frac{2M}{L}\sum_{n=2}^{\infty}e^{-D\pi^2(n^2-1)t/L^2}\longrightarrow0.
$$
The positive first profile thus controls the long-time behavior. We recover exactly the threshold $\boxed{\gamma_c=D\pi^2/L^2}$, with decay below it and unbounded positive growth at every interior point above it. The finite-mass condition is the usual meaning of an initial density profile here; standard positive-time smoothing gives the same conclusion in the usual broader admissible data classes.
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