L p − Lq estimates for solutions to the plate equation with mass term

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 − Lq estimates for the solutions in the full range 1 ≤ p ≤ q ≤ ∞, 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|α. Assuming small initial data in H2(Rn)× L2(Rn), the presence of the mass term allows us to obtain global in time existence of energy solutions for all 1 < α ≤ (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 α > (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(Rn), with p ∈ [1, 2). More precisely, we prove that for α > 1 + 4p n the solutions to the semilinear problem have the same long-time behaviour as the solutions to the linear problem.