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In this work, we give positive answer to the open question raised in [E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Maths. Pures Appl., 70 (1991) 513-529] on the exponential decay of solutions for the semilinear plate equation \[u_{tt} + \Delta^2u + a(x)u_t + \alpha u + f(u) = 0\] with localized damping. Exponential decay of the energy for this problem means that for the weak solution of this problem there exist some constants $C>1$ and $γ > 0$ such that $E(t) \leq CE(0)e^{-γt}$ , where $E(t)$ is the energy functional.