Scattering for Nonlinear Schrödinger Equations with Localized Damping in Trapping Regions

par Boris Shakarov

mardi 17 mars 2026, 11:00 → 12:00 Europe/Paris

We study the long-time behavior of nonlinear Schrödinger equations in the presence of trapping mechanisms and localized damping. Trapping typically obstructs dispersive estimates and may prevent scattering, as energy can remain concentrated along trapped trajectories. We show that suitably placed damping can counteract these effects.

A major difficulty is that the presence of inhomogeneous damping disrupts the standard energy methods used to control the dynamics. In particular, the energy is no longer monotone, and uniform bounds on the $H^1$ norm of the solution are not available a priori. To address this issue, we develop a modified energy approach inspired by virial identities, which simultaneously provides uniform control of the energy and local energy decay.

We consider two different scenarios. First, we study the cubic defocusing nonlinear Schrödinger equation with variable coefficients, allowing for strong geometric trapping. We introduce a compactly supported linear damping localized in the trapping region and prove global existence for $H^{1+\varepsilon}$ initial data, chosen because the local-in-time theory is a priori no better than for 3D unbounded manifolds. The presence of trapped trajectories leads to a loss of smoothing, preventing scattering in $H^1$, but we obtain scattering in $H^s$ for any $0 \le s < 1$.

% We then study a defocusing nonlinear Schrödinger equation with a nonlinear trapping potential and spatially dependent nonlinear damping. When the damping acts where the potential produces concentration effects, we prove global existence and scattering in the intercritical regimes, both for a damping decaying at infinity and for the asymptotically flat one.