A new critical curve for systems of evolution equations with semilinear coupling

In this work, we consider a two by two system of evolution equations with a semilinear coupling. We provide sufficient conditions on the power nonlinearities in the coupling term which guarantee the existence of global-in-time small data solutions. In particular, the coupling powers’ interplay is described by two different critical curves. We show the optimality of both critical curves, providing counterpart nonexistence results. One of the curves is analogous to the critical curve of similar evolution systems, while the second one is new for these models. This latter curve also appears if we couple a linear equation and a semilinear equation, under suitable initial data assumption. We also consider the case in which a noneffective damping is applied to the system.