Global existence and scattering for the nonlinear defocusing Schrödinger equation in 2 dimensions are known for domains exterior to star-shaped obstacles and for nonlinearities that grow at least as the quintic power. In this paper, we extend the global existence result for all non-trapping obstacles and for nonlinearities with power strictly greater than quartic. For such nonlinearities, we also prove scattering for a class of so-called almost star-shaped obstacles.
We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent p = 6 when N = 2, which does not seem to have played a role for this equation in the past. Moreover, our findings are "quite surpris…
We study the dynamics of the focusing $3d$ cubic nonlinear Schrödinger equation in the exterior of a strictly convex obstacle at the mass-energy threshold, namely, when $ E_Ω[u_0] M_Ω[u_0] = E_{\R^3}[Q] M_{\R^3}[Q] $ and $ \left\| \nabla u_0 \right\|_{L^{2}(Ω)} \left\|u_0\right\|_{L^{2}(Ω)}< \left\| \nabla Q \right\|_{L^2(\R^3)} \left\| Q \right\|_{L^2(\R^3)} ,$ where $u_0 \in H^1_0(Ω)$ is the initial data, $Q$ is the ground state on the Euclidean space, $E$ is the energy and $M$ is the mass. In the whole Euclidean space Duyckaerts and Roudenko (following the work of Duyckaerts and Merle on th…
For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that a) the finite-element method for computing approximations to solutions of the Helmholtz…
Scattering for Defocusing NLS with Inhomogeneous Nonlinear Damping and Nonlinear Trapping Potential
CNRS and Institut de Mathématiques de Toulouse（法国国家科学研究中心和图卢兹数学研究所；UMR5219，图卢兹大学；法国国家科学研究中心，图卢兹第三国立高等理工学院）；UMR5219, Université de Toulouse；CNRS, UPS IMT
Scattering in the energy space for the NLS with variable coefficients
我们考虑在$\mathbb{R}^{n}$或更一般地在具有Dirichlet边界条件的外部区域上的$n\ge3$ \begin{equation*} i \partial_t u - Lu +f(u)=0, \qquad Lv=\nabla^{b}\cdot(a(x)\nabla^{b}v)-c(x)v, \qquad \nabla^{b}=\nabla+ib(x), \end{equation*}中具有变系数的NLS，对于规范不变的幂型散焦非线性$f(u)\simeq|u|^{γ-1}u$。我们假设$L$是$Δ$的一个小的、长范围的扰动，加上一个具有大的正部分的势。本文的第一个主要结果是双线性平滑（相互作用Morawetz）估计的解决方案。作为应用，在线性流$e^{itL}$的Ehartz估计成立的条件下，证明了次临界幂$γ<1+\frac{4}{n-2}$和散射提供的$γ>1+\frac4n$在能量空间的整体适定性.当区域为$\mathbb{R}^{n}$时，通过推广Tataru [Tataru08]的Escherichartz估计，我们证明了条件假设是满足的，并推导出了能量空间中的适定性和散射性.
The purpose of the present paper is to establish appropriate cut-off resolvent estimates for the Dirichlet Laplacian on exterior domains. The geometrical assumptions on domains are rather general, for example, non-trapping condition is not imposed. The first key assumption guarantees the result on propagation of singularities, the second is the smallness of the Lebesgue measure of the portions of the trapped sets in the fibers of cosphere bundle, and the third concerns the upper bound of the sojourn time. As a by-product of these estimates, the local energy decay estimate for solutions to the …
The scattering phase, defined as $ \log \det S ( λ) / 2πi $ where $ S ( λ) $ is the (unitary) scattering matrix, is the analogue of the counting function for eigenvalues when dealing with exterior domains and is closely related to Krein's spectral shift function. We revisit classical results on asymptotics of the scattering phase and point out that it is never monotone in the case of strong trapping of waves. Perhaps more importantly, we provide the first numerical calculations of scattering phases for non-radial scatterers. They show that the asymptotic Weyl law is accurate even at low freque…
We present a numerical study of solutions to the $2d$ cubic and quintic focusing nonlinear Schrödinger equation in the exterior of a smooth, compact and strictly convex obstacle (a disk) with Dirichlet boundary condition. We first investigate the effect of the obstacle on the behavior of solutions traveling toward the obstacle at different angles and with different velocities directions. We introduce a new concept of weak and strong interactions of the solutions with the obstacle. Next, we study the existence of blow-up solutions depending on the type of the interaction and show how the presen…
