2015/02/03 by Biagio Cassano, Cassano, Biagio, Piero DʼAncona +1
Mathematics · #35L70 #58J45 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1502.00937
openalex publication_date 2015/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the NLS with variable coefficients in dimension n≥3 i ∂t u - Lu +f(u)=0, Lv=∇b⋅(a(x)∇bv)-c(x)v, ∇b=∇+ib(x), on ℝn or more generally on an exterior domain with Dirichlet boundary conditions, for a gauge invariant, defocusing nonlinearity of power type f(u)≃|u|γ-1u. We assume that L is a small, long range perturbation of Δ, 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 eitL, we prove global well posedness in the energy space for subcritical powers γ<1+(4)/(n-2), and scattering provided γ>1+\frac4n. When the domain is ℝn, by extending the Strichartz estimates due to Tataru [Tataru08], we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space.