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 \(1\le p\le q\le \infty \) , 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 \(|u|^\alpha \) . Assuming small 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 of energy solutions for all \(1<\alpha \le (n+4)/[n-4]_+\) . We 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 \(\alpha > (n+4)/[n-4]_+\) . Moreover, we study the influence on the long-time behaviour of solutions under the additional assumption of data in \(L^p(\mathbb {R}^n)\) , with \(p\in [1, 2)\) . More precisely, we prove that for \(\alpha >1 + \frac{4p}{n}\) the solutions to the semilinear problem have the same long-time behaviour as the solutions to the linear problem.
