We consider the fractional nonlinear Schrödinger equation (FNLS) with general dispersion $|\nabla|^α$ and focusing energy-critical nonlinearities $-|u|^\frac{2α}{d-α}u$ and $-(|x|^{-2α} * |u|^2) u$. By adopting Kenig-Tsutsumi \cite{mets}, Kenig-Merle \cite{keme} and Killip-Visan \cite{kv} arguments, we show the energy concentration of radial solutions near the maximal existence time. For this purpose we use Sobolev inequalities for radial functions and establish strong energy decoupling of profiles. And we also show that when the kinetic energy is confined the maximal existence time is finite for some large class of initial data satisfying the initial energy $E(φ)$ is less than energy of ground state $E(W_α)$ but $\||\nabla|^\frac\alpha2 φ\|_{L^2} \ge \||\nabla|^\frac\alpha2 W_α\|_{L^2}$.
In this paper, we use the $\bar{\partial}$ steepest descent method to study the initial value problem for focusing nonlinear Schrödinger (fNLS) equation with non-generic weighted Sobolev initial data that allows for the presence of high-order discrete spectrum. More precisely, we shall characterize the properties of the eigenfunctions and scattering coefficients in the presence of high-order poles; further we formulate an appropriate enlarged RH problem; after a series of deformations, the RH problem is transformed into a solvable model. Finally, we obtain the asymptotic expansion of the solut…
We consider the focusing modified Zakharov-Kuznetsov (mZK) equation in two space dimensions. We prove that solutions which blow up in finite time in the $H^1(\R^{2})$ norm have the property that they concentrate a non-trivial portion of their mass (more precisely, at least the amount equal to the mass of the ground state) at blow-up time. For finite-time blow-up solutions in the $H^s(\R^2)$ norm for $\frac{17}{18} < s < 1$, we prove a slightly weaker result. Moreover, we prove that the stronger concentration result can be extended to the range $ \frac{17}{18} < s \le 1$ under an additional ass…
We consider singularities of the focusing subconformal nonlinear wave equation and some generalizations of it. At noncharacteristic points on the singularity surface, Merle and Zaag have identified the rate of blow-up of the $H^1$-norm of the solution inside cones that terminate at the singularity. We derive bounds that restrict how this $H^1$-energy can be distributed inside such cones. Our proof relies on new localized estimates obtained using Carleman- type inequalities for such nonlinear waves. These bound the $L^{p+1}$-norm in the interior of timelike cones by their $H^1$-norm near the bo…
In this paper, we prove a refined version of a compactness lemma and we use it to establish mass-concentration for the focusing nonlinear Schrödinger equation with an inverse-square potential.
In this paper we study dynamical properties of blowup solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. We establish a profile decomposition and a compactness lemma related to the equation. As a result, we obtain the $L^2$-concentration and the limiting profile with minimal mass of blowup solutions.
We revisit the scattering result of Holmer and Roudenko on the radial focusing cubic NLS in three space dimensions. Using the radial Sobolev embedding and a virial/Morawetz estimate, we give a simple proof of scattering below the ground state that avoids the use of concentration compactness.
We revisit the scattering result of Duyckaerts, Holmer, and Roudenko for the non-radial $\dot H^{1/2}$-critical focusing NLS. By proving an interaction Morawetz inequality, we give a simple proof of scattering below the ground state in dimensions $d\geq 3$ that avoids the use of concentration compactness.
We consider the scattering results of the radial solutions below the ground state to the focusing inhomogeneous nonlinear Schrödinger equation $$i\partial_tu+Δu +|x|^{-b}|u|^{p}u=0$$ in two dimension, where $0<b<1$ and $2-b<p<\infty$. We use a modified version of Arora-Dodson-Murphy's approach [1] to give a new proof that extends the scattering results of [10] and avoids concentration compactness.
In this paper, we consider the scattering theory of the radial solution to focusing energy-subcritical Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$ using the method from \cite{Dodson2016}. The main difficulties are the equation is \emph{not} space-translation invariant and the nonlinearity is non-local. Using the radial Sobolev embedding and a virial-Morawetz type estimate we can exclude the concentration of mass near the origin. Besides, we can overcome the weak dispersive estimate when $a<0$, using the dispersive estimate established by \cite{zheng…
The aim of this note is to adapt the strategy in [4][See,B.Dodson, J.Murphy, a new proof of scattering below the ground state for the 3D radial focusing cubic NLS, arXiv:1611.04195 ] to prove the scattering of radial solutions below sharp threshold for certain focusing fractional NLS with cubic nonlinearity. The main ingredient is to apply the fractional virial identity proved in [11][See,T.Boulenger, D.Himmelsbach,E.Lenzmann, Blow up for fractional NLS,J.Func.Anal,271(2016),2569-2603] to exclude the concentration of mass near the origin.
We study solutions of the focusing energy-critical nonlinear heat equation $u_t = Δu - |u|^2u$ in $\mathbb{R}^4.$ We show that solutions emanating from initial data with energy and $\dot{H}^1-$norm below those of the stationary solution $W$ are global and decay to zero, via the "concentration-compactness plus rigidity" strategy of Kenig-Merle. First, such global solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza, Seregin and Sverak in an argu…
In this paper, we consider the dynamics of the solution to the mass critical focusing Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$. The main difficulties are the equation is \emph{not} space-translation invariant and the nonlinearity is non-local. We first prove that if the mass of the initial data is less than that of ground states, then the solution will be global. Although we don't know whether the ground state is unique, we can verify all the ground states have the same, minimal mass threshold. Then at the minimal mass threshold, we can construct…
We study dynamical properties of blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation \[ i\partial_t u -(-Δ)^s u = -|u|^αu, \quad u(0) = u_0, \quad \text{on } [0,\infty) \times \mathbb{R}^d, \] where $d \geq 2, \frac{d}{2d-1} \leq s <1$, $\frac{4s}{d}<α<\frac{4s}{d-2s}$ and $u_0 \in \dot{H}^{s_{\text{c}}} \cap \dot{H}^s$ is radial with the critical Sobolev exponent $s_{\text{c}}$. To this end, we establish a compactness lemma related to the equation by means of the profile decomposition for bounded sequences in $\dot{H}^{s_{\text{c}}} \cap \dot{H}^s$. …
We consider the focusing inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ $$i \partial_t u +Δu + |x|^{-b} |u|^{2σ}u = 0,$$ where $N\geq 2$ and $σ$, $b>0$. We first obtain a small data global result in $H^1$, which, in the two spatial dimensional case, improves the third author result in [22] on the range of $b$. For $N\geq 3$ and $\frac{2-b}{N}<σ<\frac{2-b}{N-2}$, we also study the local well posedness in $\dot H^{s_c}\cap \dot H^1 $, where $s_c=\frac{N}{2}-\frac{2-b}{2σ}$. Sufficient conditions for global existence of solutions in $\dot H^{s_c}\cap \dot H^1$ are also esta…
We consider the following class of focusing $L^2$-supercritical fourth-order nonlinear Schrödinger equations \[ i\partial_t u - Δ^2 u + μΔu = - |u|^αu, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^N, \] where $N\geq 2$, $μ\geq 0$, and $\frac{8}{N}<α<α^*$ with $α^*:=\frac{8}{N-4}$ if $N\geq 5$ and $α^*=\infty$ if $N\leq 4$. By using the localized Morawetz estimates and radial Sobolev embedding, we establish the energy scattering below the ground state threshold for the equation with radially symmetric initial data. We also address the existence of finite time blow-up radial solutions to the equ…
We consider the inhomogeneous nonlinear Schrödinger equation $$ i u_t +Δu+|x|^{-b}|u|^αu = 0, $$ where $\frac{4-2b}{N}<α<\frac{4-2b}{N-2}$ (when $N=2$, $\frac{4-2b}{N}<α<\infty$) and $0<b<\min\{N/3,1\}$. For a radial initial data $u_0\in H^1(\mathbb{R}^N)$ under a certain smallness condition we prove that the corresponding solution is global and scatters. The smallness condition is related to the ground state solution of $-Q+ΔQ+ |x|^{-b}|Q|^αQ=0$ and the critical Sobolev index $s_c=\frac{N}{2}-\frac{2-b}α$. This is an extension of the recent work \cite{paper2} by the same authors, where they c…
Scattering below the ground state for the three-dimensional focusing energy-critical NLS
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Morningside Center of Mathematics（晨兴数学中心）；School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
We study the focusing intercritical NLS \begin{align}\label{abstract_nls} i\pt_t u+Δ_{x,y}u=-|u|^αu\tag{NLS} \end{align} on the semiperiodic waveguide manifold $\R^d_x\times \T_y$ with $d\geq 5$ and $α\in(\frac{4}{d},\frac{4}{d-1})$. In the case $d\leq 4$, with the aid of the semivirial vanishing theory \cite{Luo_inter}, the author was able to construct a sharp threshold, which being uniquely characterized by the ground state solutions, that sharply determines the bifurcation of global scattering and finite time blow-up solutions in dependence of the sign of the semivirial functional. As the d…
We consider the focusing nonlinear Schrödinger equation $i u_t + Δu + |u|^{p-1}u=0$, $p>1,$ and the generalized Hartree equation $iv_t + Δv + (|x|^{-(N-γ)}\ast |v|^p)|v|^{p-2}u=0$, $p\geq2$, $γ<N$, in the mass-supercritical and energy-subcritical setting. With the initial data $u_0\in H^1(\mathbb{R}^N)$ the characterization of solutions behavior under the mass-energy threshold is known for the NLS case from the works of Holmer and Roudenko in the radial [16] and Duyckaerts, Holmer and Roudenko in the nonradial setting [10] and further generalizations (see [1,11,14]); for the generalized Hartre…
Finite-mass soliton-type rigidity and four-channel reduction for the three-dimensional nonradial focusing energy-critical nonlinear Schrödinger equation
Guangdong University of Education（广东教育学院）；University of Louisville（路易斯维尔大学）
We consider the semilinear wave equation with focusing energy-critical nonlinearity in space dimension 5 with radial data. It is known that a solution $(u, \partial_t u)$ which blows up at $t = 0$ in a neighborhood (in the energy norm) of the family of solitons $W_λ$, asymptotically decomposes in the energy space as a sum of a bubble $W_λ$ and an asymptotic profile $(u_0^*, u_1^*)$, where $\lim_{t\to 0}λ(t)/t = 0$ and $(u^*_0, u^*_1) \in \dot H^1\times L^2$. We construct a blow-up solution of this type such that $(u^*_0, u^*_1)$ is any pair of sufficiently regular functions with $u_0^*(0) > 0$…
We prove the existence of a global solution of the energy-critical focusing wave equation in dimension $5$ blowing up in infinite time at any $K$ given points $z_k$ of $\mathbb{R}^5$, where $K\geq 2$. The concentration rate of each bubble is asymptotic to $c_k t^{-2}$ as $t\to \infty$, where the $c_k$ are positive constants depending on the distances between the blow-up points $z_k$. This result complements previous constructions of blow-up solutions and multi-solitons of the energy-critical wave equation in various dimensions $N\geq 3$.
We show that the solutions of the three-dimensional critical defocusing nonlinear wave equation with Neumann boundary conditions outside a ball and radial initial data scatter. This is to our knowledge the first result of scattering for a nonlinear wave equation with Neumann boundary conditions. Our proof uses the scheme of concentration-compactness/rigidity introduced by Kenig and Merle, extending it to our setup, together with the so-called channels of energy method to rule out compact-flow solutions. We also obtain, for the focusing equation, the same exact scattering/blow-up dichotomy belo…
In this paper we consider the real-valued mass-critical nonlinear Klein-Gordon equations in three and higher dimensions. We prove the dichotomy between scattering and blow-up below the ground state energy in the focusing case, and the energy scattering in the defocusing case. We use the concentration-compactness/rigidity method as R. Killip, B. Stovall, and M. Visan [Trans. Amer. Math. Soc. 364 (2012)]. The main new novelty is to approximate the large scale (low-frequency) profile by the solution of the mass-critical nonlinear Schrödinger equation when the nonlinearity is not algebraic.
Global weak solution of 3-D focusing energy-critical nonlinear Schrödinger equation
Hohai University（河海大学）；Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学交叉前沿研究中心）；Frontiers Science Center for Nonlinear Expectations, Ministry of Education（教育部非线性期望前沿科学中心）；Department of Physics and Mathematics, University of Eastern Finland（东芬兰大学物理与数学系）；School of Mathematics, Shandong University（山东大学数学学院）
