Solution
= Solution
Each spatial mode acquires the factor
$$
\exp\left(\int_0^t\mu(s)\,ds-Dk_n^2t\right)
=\exp\left((\mu_0-Dk_n^2)t+\mu_1\sin t\right).
$$
The oscillatory factor $e^{\mu_1\sin t}$ is bounded and does not affect the long-term exponential rate. With two open ends, growth therefore occurs exactly when
$$
\mu_0>\frac{D\pi^2}{L^2}.
$$
Only the time average of this spatially uniform fluctuating growth rate controls the asymptotic threshold.