Abstract: The purpose of this talk is to explain the decay problem for the generalized improved (or regularized) Boussinesq model with power type nonlinearity, a modification of the originally ill-posed shallow water waves model derived by Boussinesq.
The associated decay problem has been studied by Liu, and more recently by Cho-Ozawa, showing scattering in weighted spaces provided the power of the nonlinearity $p$ is sufficiently large. In this talk we remove that condition on the power $p$ and prove decay to zero in terms of the energy space norm $L^2\times H^1$, for any $p>1$, in two almost complementary regimes: (i) outside the light cone for all small, bounded in time $H^1\times H^2$ solutions, and (ii) decay on compact sets of arbitrarily large bounded in time $H^1\times H^2$ solutions.
These works were developed in collaboration with C. Muñoz.