= Solution
Write
$$
w'=\sum_l i\pi l\,\widehat w_l e^{i\pi lx},
\qquad
u_x=\sum_m i\pi m\,\widehat u_m e^{i\pi mx}.
$$
Projecting the equation onto modes $|n|\leq D$ gives the <Fourier-Galerkin matrix for a drift-diffusion equation>
$$
\boxed{\dot{\widehat u}_n
=\sum_{|m|\leq D}B_{nm}\widehat u_m,\qquad
B_{nm}=-\pi^2n^2\delta_{nm}
+\pi^2(n-m)m\,\widehat w_{n-m},}
$$
where $\widehat w_l=0$ for $|l|>d$.
For $w=\cos\pi x$, only
$\widehat w_{\pm1}=1/2$ are nonzero. Hence
$$
B_{nm}=-\pi^2n^2\delta_{nm}
+\frac{\pi^2}{2}(n-m)m\,\mathbf1_{\{|n-m|=1\}}.
$$
The column belonging to the constant mode $m=0$ is zero, so
$$
\boxed{B\text{ is not invertible}}.
$$
Remove that zero mode. In every remaining row $n$, the sum of the magnitudes of the off-diagonal entries is at most $\pi^2|n|$; for $|n|=1$, the would-be coupling to the removed zero mode vanishes. The diagonal entry is $-\pi^2n^2$. Gershgorin's theorem therefore places every <eigenvalue> of the nonconstant block in
$$
\operatorname{Re}z\leq-\pi^2(n^2-|n|)\leq0.
$$
Together with the zero <eigenvalue>, all <eigenvalues> of $B$ have nonpositive real part.
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