Scattering in the energy space for the NLS with variable coefficients. (English) Zbl 1352.35158
Summary: We consider the NLS with variable coefficients in dimension \(n\geq 3\)
\[
i\partial_tu-Lu+f(u)=0,\quad Lv=\nabla^b\cdot(a(x)\nabla^bv)-c(x)v,\qquad \nabla^b=\nabla +ib(x),
\]
on \(\mathbb R^n\) or more generally on an exterior domain with Dirichlet boundary conditions, for a gauge invariant, defocusing nonlinearity of power type \(f(u)\simeq |u|^{\gamma-1}u\). We assume that \(L\) is a small, long range perturbation of \(\Delta\), plus a potential with a large positive part. The first main result of the paper is a bilinear smoothing (interaction Morawetz) estimate for the solution. As an application, under the conditional assumption that Strichartz estimates are valid for the linear flow \(e^{itL}\), we prove global well posedness in the energy space for subcritical powers \(\gamma<1+\frac{4}{n-2}\), and scattering provided \(\gamma>1+\frac{4}{n}\). When the domain is \(\mathbb R^n\), by extending the Strichartz estimates due to D. Tataru [Am. J. Math. 130, No. 3, 571–634 (2008; Zbl 1159.35315)], we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35L70 | Second-order nonlinear hyperbolic equations |
| 58J45 | Hyperbolic equations on manifolds |
| 35P25 | Scattering theory for PDEs |
| 35L71 | Second-order semilinear hyperbolic equations |
| 35B45 | A priori estimates in context of PDEs |
Citations:
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