= Solution
Set $h_x=1/(N+1)$ and $J=2N+1$. The <Dirichlet discrete Laplacian> has the <eigenvectors>
$$
v_j(m)=\sin\frac{j\pi(m+N+1)}{2N+2},\qquad j=1,\ldots,2N+1,
$$
and its negative has the <eigenvalues>
$$
\lambda_j^h=\frac4{h_x^2}\sin^2\frac{j\pi}{4(N+1)}.
$$
The <discrete sine transform> therefore reduces the <method of lines> equations to $q_j''=(\alpha-\lambda_j^h)q_j$. Each coefficient is an oscillator, a linear function, or a hyperbolic function according to the sign of $\lambda_j^h-\alpha$. A finite collection of oscillators is bounded in every fixed-grid <norm>. A zero-frequency mode can grow linearly, and a negative-frequency-square mode can grow exponentially. Consequently,
$$
\boxed{\alpha<4(N+1)^2\sin^2\frac{\pi}{4(N+1)}}
$$
is necessary and sufficient for <all-time boundedness of a semidiscrete reaction wave equation> on this grid. Equality allows bounded displacement only when the initial velocity has zero projection onto the first <eigenvector>; above it, the growing components must be absent in every unstable mode. A strictly positive <spectral gap> also gives a time-independent displacement bound uniform over grids whose gaps are bounded below.
The discrete threshold is strictly smaller than $\pi^2/4$, because $\sin s<s$ for $s>0$, and tends to $\pi^2/4$ as $N\to\infty$. A coarse grid can therefore introduce long-time growth into a continuum problem with $\alpha<\pi^2/4$. This is distinct from <displacement stability of a symmetric semidiscrete wave equation> on fixed finite time intervals. As in part (a), if existence of the printed limit is required for every initial datum, \b[no $\alpha$ qualifies]: any one mode admits oscillatory, secular, or growing initial data, depending on its coefficient.
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