= Solution
Use the <Dirichlet Laplacian eigenvalues> and <eigenfunctions> on $(-1,1)$:
$$
e_j(x)=\sin\frac{j\pi(x+1)}2,\qquad \lambda_j=\left(\frac{j\pi}2\right)^2,\qquad j\geq1.
$$
An expansion $u=\sum_jq_j(t)e_j$ reduces the <wave equation> to independent <ordinary differential equations>
$$
q_j''=(\alpha-\lambda_j)q_j.
$$
When $\alpha<\lambda_1$, put $\omega_j=\sqrt{\lambda_j-\alpha}$; each coefficient is
$$
q_j(t)=\phi_j\cos(\omega_jt)+\frac{\psi_j}{\omega_j}\sin(\omega_jt).
$$
For finite-energy data $\phi\in H_0^1(-1,1)$ and $\psi\in L^2(-1,1)$, a uniform spatial bound follows from the <energy method>, rather than from unjustified absolute summation of the <Fourier series>. The conserved quantity is
$$
E=\frac12\left(\|u_t\|_2^2+\|u_x\|_2^2-\alpha\|u\|_2^2\right).
$$
The <Poincare inequality> gives $\|u\|_2^2\leq\lambda_1^{-1}\|u_x\|_2^2$. Hence $2E\geq c_\alpha\|u_x\|_2^2$ with $c_\alpha=1-\alpha/\lambda_1>0$ for $0\leq\alpha<\lambda_1$, and with $c_\alpha=1$ for $\alpha<0$. Since the <Dirichlet boundary condition> gives $u(x,t)=\int_{-1}^xu_x(s,t)\,ds$, the <Cauchy-Schwarz inequality> yields
$$
\sup_{t\geq0}\|u(\cdot,t)\|_\infty\leq 2\sqrt{E/c_\alpha}<\infty.
$$
At $\alpha=\lambda_1$, the first coefficient is $q_1(t)=\phi_1+t\psi_1$, which is unbounded for some admissible data. For $\alpha>\lambda_1$, that coefficient has an exponentially growing component for some admissible data. Thus <all-time boundedness of a wave equation with a reaction term> requires
$$
\boxed{\alpha<\pi^2/4.}
$$
At equality, boundedness for a particular data set requires $\psi_1=0$; the remaining modes are bounded by their <spectral gap>. Above the threshold, every unstable mode must have its growing component canceled, and any zero-frequency mode must have zero initial velocity. There is no condition on $\alpha$ alone for arbitrary specially chosen data.
The printed reference to a limit needs qualification. Bounded oscillations generally have no limit as $t\to\infty$. If actual existence of that limit for every admissible initial datum is required, \b[no real $\alpha$ works]: for every $\alpha$, choose $j$ with $\lambda_j>\alpha$ and nonzero pure oscillatory data in that <eigenfunction>. The boxed inequality answers the intended long-time boundedness question.
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