= Solution
The <Dirichlet Laplacian eigenfunctions> and positive <Dirichlet Laplacian eigenvalues> on the interval of length two are
$$
\phi_j(x)=\sin\frac{j\pi(x+1)}2,\qquad
\lambda_j=\left(\frac{j\pi}{2}\right)^2,\qquad j\geq1.
$$
A <Sturm-Liouville eigenfunction expansion> gives $u=\sum_jq_j(t)\phi_j$, with $q_j''+(\lambda_j-\alpha)q_j=0$. Thus all <normal modes> oscillate precisely when $\alpha<\lambda_1$.
To justify a spatially uniform bound for arbitrary appropriate data, assume the usual <wave equation> energy class $u(0)\in H_0^1(-1,1)$ and $u_t(0)\in L^2(-1,1)$, or smoother compatible data. The <energy method> conserves
$$
E=\frac12\left(\|u_t\|_{L^2}^2+\|u_x\|_{L^2}^2-\alpha\|u\|_{L^2}^2\right).
$$
The sharp <Poincaré inequality> is $\|u\|_{L^2}^2\leq\lambda_1^{-1}\|u_x\|_{L^2}^2$. For $0\leq\alpha<\lambda_1$, this makes $E$ coercive with constant $1-\alpha/\lambda_1$; for $\alpha<0$, <coercivity> is immediate. The <one-dimensional Sobolev representative> satisfies $|u(x,t)|\leq\sqrt2\,\|u_x(t)\|_{L^2}$ by the <fundamental theorem of calculus> and the <Cauchy-Schwarz inequality>. Hence the conserved <energy> bounds $u$ uniformly in both variables.
At $\alpha=\lambda_1$, the smooth solution $u=t\phi_1$ is unbounded. At $\alpha>\lambda_1$, $u=e^{\sqrt{\alpha-\lambda_1}t}\phi_1$ is unbounded. Therefore
$$
\boxed{\alpha<\frac{\pi^2}{4}}.
$$
The bound depends on the initial data; it is not a single bound for all arbitrarily rescaled solutions. The energy-class hypothesis supplies pointwise meaning and excludes undefined rough-data interpretations.
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