# Decay in the one dimensional generalized Improved Boussinesq equation

## TL;DR

This paper proves decay to zero of solutions to the one-dimensional generalized Improved Boussinesq equation for any nonlinearity power p>1, removing previous restrictions, using new virial estimates in different spatial regimes.

## Contribution

It introduces novel virial identities to establish decay in the energy space for all p>1, extending prior results that required larger p values.

## Key findings

- Decay to zero in energy space for all p>1
- Decay outside the light cone for small solutions
- Decay on large bounded sets for solutions over time

## Abstract

We consider 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. This equation has been extensively studied in the literature, describing plenty of interesting behavior, such as global existence in the space $H^1\times H^2$, existence of super luminal solitons, and lack of a standard stability method to describe perturbations of solitons. 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 paper 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. The proof consists in finding two new virial type estimates, one for the exterior cone problem based in the energy of the solution, and a more subtle virial identity for the interior cone problem, based in a modification of the momentum.