  In this paper, we study the Cauchy problem for the linear plate equation with
mass term and its applications to semilinear models. For the linear problem we
obtain $L^p-L^q$ estimates for the solutions in the full range $1\leq p\leq
q\leq \infty$, and we show that such estimates are optimal. In the sequel, we
discuss the global in time existence of solutions to the associated semilinear
problem with power nonlinearity $|u|^\alpha$. For low dimension space $n\leq
4$, and assuming $L^1$ regularity on the second datum, we were able to prove
global existence for $\alpha> \max\{\alpha_c(n), \tilde\alpha_c(n)\}$ where
$\alpha_c = 1+4/n$ and $\tilde \alpha_c = 2+2/n$. However, assuming initial
data in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$, the presence of the mass
term allows us to obtain global in time existence for all $1<\alpha \leq
(n+4)/[n-4]_+$. We also show that the latter upper bound is optimal, since we
prove that there exist data such that a non-existence result for local weak
solutions holds when $\alpha > (n+4)/[n-4]_+$.
