The paper deals with the defocusing case of the energy subcritical non-linear wave equation in $R^3$. We assume the initial data is in the space $\dot{H}^s \times \dot{H}^{s-1}$ and radial. If $s=1$, this is the energy space and the scattering results are known. In this paper we choose $15/16<s<1$ and assume that the nonlinearity equals $-|u|^p u$ with $p = 2/{3/2 -s}$, so that this problem is $\dot{H}^s \times \dot{H}^{s-1}$ critical. We will prove the global well-posedness and scattering of the solution under the additional assumption that the $\dot{H}^s \times \dot{H}^{s-1}$ norm of the solution is uniformly bounded for all time $t$ in the maximal lifespan of the solution.
We consider the Cauchy problem for the defocusing power type nonlinear wave equation in $(1+3)$-dimensions for energy subcritical powers $p$ in the range $3 < p< 5$. We prove that any solution is global-in-time and scatters to free waves in both time directions as long as its critical Sobolev norm stays bounded on the maximal interval of existence.
In this paper the author considers the global existence and well-posedness of the non-linear wave equation $\partial_t^2 u - Δu = -|u|^{p-1} u$ in 3-dimensional space, assuming that the initial data is in the space $(\dot{H}^s \cap \dot{H}^{s_p}) \times (\dot{H}^{s-1} \cap \dot{H}^{s_p-1})$, with $11/3<p<5$, $s_p = 3/2 - 2/(p-1)$ and $s$ sufficiently close to 1. The main idea is to use the I-method and consider the almost conservation law of the energy.
We show robustness of various truncated and subcritical approximations to the stochastic defocusing mass-critical nonlinear Schrödinger equation (NLS) in dimension $d=1$, whose solution was constructed in [FX18] with one particular such approximation. The key ingredient in the proof is a uniform bound of the solutions to the family of deterministic mass-subcritical defocusing NLS.
We prove the scattering for the defocusing generalized Benjamin-Ono equation in the energy space $H^{\frac{1}{2}}(\mathbb{R})$. We first establish the monotonicity formula that describes the unidirectional propagation. More precisely, it says that the center of energy moves faster than the center of mass. This type of monotonicity was first observed by Tao (DCDS 2007) in the defocusing gKdV equations. We use the monotonicity in the setting of compactness-contradiction argument to prove the large data scattering in the energy space $H^{\frac{1}{2}}(\mathbb{R})$. On the way, we extend critical l…
In this paper we continue the study of the defocusing, energy-subcritical nonlinear wave equation with radial initial data lying in the critical Sobolev space. In this case we prove scattering in the critical norm when $3 < p < 5$.
Large-data modified wave operators for the defocusing nonlinear Schrödinger equation in one space dimension with subcritical long-range nonlinearity
Research Institute for Interdisciplinary Science, Okayama University（冈山大学跨学科科学研究所）；Department of Mathematics, Graduate School of Science, The University of Osaka（大阪大学理学研究科数学系）
In this paper we prove global well-posedness for the defocusing, energy-subcritical, nonlinear wave equation on $\mathbb{R}^{1 + 3}$ with initial data in a critical Besov space. No radial symmetry assumption is needed.
In this paper we prove global well-posedness and scattering for the defocusing, cubic, nonlinear wave equation on $\mathbf{R}^{1 + 3}$ with radial initial data lying in the critical Sobolev space $\dot{H}^{1/2}(\mathbf{R}^{3}) \times \dot{H}^{-1/2}(\mathbf{R}^{3})$.
We consider the radial defocusing nonlinear Schrödinger equations $iu_t+Δu=|u|^{p}u$ with supercritical exponent $p>4$ in four space dimensions, and prove that any radial solution that remains bounded in the critical Sobolev space must be global and scatter.
In this paper we prove that the cubic wave equation is globally well - posed and scattering for radial initial data lying in a slightly supercritical Sobolev space, and a weighted Sobolev space.
We extend the scattering result for the radial defocusing-focusing mass-energy double critical nonlinear Schrödinger equation in $d\leq 4$ given by Cheng et al. to the case $d\geq 5$. The main ingredient is a suitable long time perturbation theory which is applicable for $d\geq 5$. The paper will therefore give a full characterization on the scattering threshold for the radial defocusing-focusing mass-energy double critical nonlinear Schrödinger equation in all dimensions $d\geq 3$.
We study the defocusing nonlinear Schrödinger equation in three space dimensions. We prove that any radial solution that remains bounded in the critical Sobolev space must be global and scatter. In the energy-supercritical setting, we employ a space-localized Lin--Strauss Morawetz inequality of Bourgain. In the inter-critical regime, we prove long-time Strichartz estimates and frequency-localized Lin--Strauss Morawetz inequalities.
We consider the mass-subcritical NLS in dimensions $d\geq 3$ with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobolev space throughout its lifespan must be global and scatter. In the focusing case, we prove the existence of a threshold solution that has a compact flow.
In this paper, we study the defocusing cubic Schrödinger equation on three dimensional hyperbolic space $\mathbb{H}^3$ with radial initial data in the Sobolev Space $H^s(0<s<1)$. Our main result is that the initial value problem is globally wellposed and scatters for $\frac{15}{16}<s<1$. This is an extension of the work of Staffilani and Yu to the three dimensional hyperbolic space.
We prove the global well-posedness and scattering for the defocusing $H^{\frac12}$-subcritical (that is, $2<γ<3$) Hartree equation with low regularity data in $\mathbb{R}^d$, $d\geq 3$. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space $H^s\big(\mathbb{R}^d\big)$ with $s>4(γ-2)/(3γ-4)$, which also scatters in both time directions. This improves the result in \cite{ChHKY}, where the global well-posedness was established for any $s>\max\big(1/2,4(γ-2)/(3γ-4)\big)$. The new ingredients in our proof are that we make use of an interaction Morawetz est…
We consider the pure-power defocusing nonlinear Klein-Gordon equation, in the energy subcritical case, posed on the product space $\mathbb R^d\times \mathbb T$, where $\mathbb T$ is the one-dimensional flat torus. In this framework, we prove that scattering holds for any initial data belonging to the energy space $H^1 \times L^2$ for $1\leq d\leq 4$. The strategy consists in proving a suitable profile decomposition theorem in $\mathbb R^d\times \mathbb T$ to pursue a concentration-compactness \& rigidity method.
We consider the mass-subcritical nonlinear Schrödinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity $s_c\in(\max\{-1,-\frac{d}{2}\},0)$, we prove that any solution satisfying $\|\, |x|^{|s_c|}e^{-itΔ} u\|_{L_t^\infty L_x^2} <\infty$ on its maximal interval of existence must be global and scatter.
In this work, we consider the energy-supercritical defocusing cubic nonlinear wave equation in dimension d=5 for radially symmetric initial data. We prove that an a priori bound in the critical space implies global well-posedness and scattering. The main tool that we use is a frequency localized version of the classical Morawetz inequality, inspired by recent developments in the study of the mass and energy critical nonlinear Schrödinger equation.
We consider the defocusing energy-critical NLS in the exterior of the unit ball in three dimensions. For the initial value problem with Dirichlet boundary condition we prove global well-posedness and scattering with large radial initial data in the Sobolev space $\dot H_0^1$. We also point out that the same strategy can be used to treat the energy-supercritical NLS in the exterior of balls with Dirichlet boundary condition and radial $\dot H_0^1$ initial data.
In this paper we prove that the cubic wave equation is globally well - posed and scattering for radial initial data lying in $B_{1,1}^{2} \times B_{1,1}^{1}$. This space of functions is a scale invariant subspace of $\dot{H}^{1/2} \times \dot{H}^{-1/2}$.
Using channel of energy inequalities developed by T.Duyckaerts, C.Kenig and F.Merle, we prove that, modulo a free radiation, any finite energy radial solution to the defocusing energy critical wave equation with radial potential in 3 + 1 dimensions converges to the set of steady states as time goes to infinity. For generic potentials we prove there are only finitely many steady states, and in this case modulo some free radiation the solution converges to one steady state as time goes to infinity.
In this paper, we consider the wave equation in 3-dimensional space with an energy-subcritical nonlinearity, either in the focusing or defocusing case. We show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined in time and scatters. The proof depends on the compactness/rigidity argument, decay estimates for radial, "compact" solutions, gain of regularity arguments and the "channel of energy" method.
We consider the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^4$ and establish almost sure global existence and scattering for randomized radially symmetric initial data in $H^s_x(\mathbb{R}^4) \times H^{s-1}_x(\mathbb{R}^4)$ for $\frac{1}{2} < s < 1$. This is the first almost sure scattering result for an energy-critical dispersive or hyperbolic equation with scaling super-critical initial data. The proof is based on the introduction of an approximate Morawetz estimate to the random data setting and new large deviation estimates for the free wave evolution of randomized ra…
In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is $\dot{H}^{1/2} \times \dot{H}^{-1/2}$. We show that if the initial data is radial and lies in $(\dot{H}^{s} \times \dot{H}^{s - 1}) \cap (\dot{H}^{1/2} \times \dot{H}^{-1/2})$ for some $s > \frac{1}{2}$, then the cubic initial value problem is globally well - posed. We use the I - method and the long time Strichartz estimates. This method is quite similar to the method used in [D2].
We consider the defocusing nonlinear wave equation $u_{tt}-Δu + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2}<p<\frac4{d-3}$ (which is energy-supercritical) and dimensions $3\leq d\leq 6$; we also consider $d\geq 7$, but for a smaller range of $p>\frac4{d-2}$. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.
We prove global pointwise decay estimates for a class of defocusing semilinear wave equations in $n=3$ dimensions restricted to spherical symmetry. The technique is based on a conformal transformation and a suitable choice of the mapping adjusted to the nonlinearity. As a result we obtain a pointwise bound on the solutions for arbitrarily large Cauchy data, provided the solutions exist globally. The decay rates are identical with those for small data and hence seem to be optimal. A generalization beyond the spherical symmetry is suggested.
We investigate the initial value problem for a defocusing nonlinear Schrödingerequation with exponential nonlinearity. We identify subcritical, critical and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.
In this paper, we investigate the global behaviors of solutions to defocusing semilinear wave equations in $\mathbb{R}^{1+d}$ with $d\geq 3$. We prove that in the energy space the solution verifies the integrated local energy decay estimates for the full range of energy subcritical and critical power. For the case when $p>1+\frac{2}{d-1}$, we derive a uniform weighted energy bound for the solution as well as inverse polynomial decay of the energy flux through hypersurfaces away from the light cone. As a consequence, the solution scatters in the energy space and in the critical Sobolev space fo…
In this paper, we consider the defocusing nonlinear Schrödinger equation in space dimensions $d\geq 4$. We prove that if $u$ is a radial solution which is \emph{priori} bounded in the critical Sobolev space, that is, $u\in L_t^\infty \dot{H}^{s_c}_x$, then $u$ is global and scatters. In practise, we use weighted Strichartz space adapted for our setting which ultimately helps us solve the problems in cases $d\geq 4$ and $0<s_c<\f{1}{2}$. The results in this paper extend the work of \cite[Comm. in PDEs, 40(2015), 265-308]{M3} to higher dimensions.
In this paper, we study the global well-posedness and scattering of 3D defocusing, cubic Schrödinger equation. Recently, Dodson [arXiv:2004.09618] studied the global well-posedness in a critical Sobolev space $\dot{W}^{11/7,7/6}$. In this paper, we aim to show that if the initial data belongs to $\dot H^\frac12$ to guarantee the local existence, then some extra weak space which is subcritical, is sufficient to prove the global well-posedness. More precisely, we prove that if the initial data belongs to $\dot{H}^{1/2}\cap \dot{W}^{s,1}$ for $12/13<s \leqslant 1$, then the corresponding solution…
By $I$-method, the interaction Morawetz estimate, long time Strichartz estimate and local smoothing effect of Schrödinger operator, we show global well-posedness and scattering for the defocusing Hartree equation $$\left\{ \begin{array}{ll} iu_t + Δu &=F(u), \quad (t,x) \in \mathbb{R} \times \mathbb{R}^4 u(0) \\ &=u_0(x)\in H^s(\mathbb{R}^4), \end{array} \right. $$ where $F(u)= (V* |u|^2) u$, and $V(x)=|x|^{-γ}$, $3< γ<4$, with radial data in $H^{s}(\mathbb{R}^4)$ for $s>s_c:=γ/2-1$. It is a sharp global result except of the critical case $s=s_c$, which is a very difficult open problem.
We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - Δu = -|u|^{p-1} u$ in 4 to 6 dimensional spaces with radial initial data. We define $w=r^{(d-1)/2} u$, reduce the equation above to one-dimensional equation of $w$ and apply method of characteristic lines. This gives scattering of solutions outside any given light cone as long as the energy is finite. The scattering in the whole space can also be proved if we assume the energy decays at a certain rate as $x\rightarrow +\infty$. This generalize the 3-dimensional results in Shen[27] to higher dimensions.
In this paper, we study the global well-posedness and scattering theory for the defocusing fourth-order nonlinear Schrödinger equation (FNLS) $iu_t+Δ^2 u+|u|^pu=0$ in dimension $d\geq9$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty(I;\dot H^{s_c}_x(\R^d))$ with all $s_c:=\frac{d}2-\frac4p\geq1$ if $p$ is an even integer or $s_c\in[1,2+p)$ otherwise, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical and energy-subcritical nonlinear Schrödinger e…
In this paper, we obtain the global well-posedness and scattering for the radial solution to the defocusing conformal invariant nonlinear wave equation with initial data in the critical Besov space $\dot{B}^3_{1,1}\times\dot{B}^2_{1,1}(\mathbb{R}^5)$. This is the five dimensional analogue of \cite{dodson-2016}, which is the first result on the global well-posedness and scattering of the energy subcritical nonlinear wave equation without the uniform boundedness assumption on the critical Sobolev norms employed as a substitute of the missing conservation law with respect to the scaling invarianc…
We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - Δu = -|u|^{p-1} u$ in dimension $d \in \{3,4,5\}$ and prove the exterior scattering of solutions if $3\leq d \leq 5$ and $1+6/d<p<1+4/(d-2)$. More precisely, given any solution with a finite energy, there exists a solution $u_L$ to the homogeneous linear wave equation, so that the following limit holds \[ \lim_{t\rightarrow +\infty} \int_{|x|>t+R} |\nabla_{x,t} u(x,t)- \nabla_{x,t} u_L(x,t)|^2 dx = 0 \] for any fixed real number $R$. This generalize the previously known exterior scattering result in the radial case.
In this paper, using $I$-method, we establish $H^s_x\times H^s_x$ scattering theories for the following Cauchy problem \begin{equation*} \left\{ \begin{array}{lll} iu_t+Δu=λ|v|^2u,\quad iv_t+Δv=μ|u|^2v,\quad x\in \mathbb{R}^3,\ t>0,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in \mathbb{R}^3. \end{array}\right. \end{equation*} Here $λ>0$, $μ>0$, $(u_0,v_0)\in H^s_x(\mathbb{R}^3)\times H^s_x(\mathbb{R}^3)$ and $\frac{1}{2}<s<1$.
We consider a perturbed energy critical focusing Nonlinear Schrödinger Equation in three dimensions. We construct solitary wave solutions for focusing subcritical perturbations as well as defocusing supercritical perturbations. The construction relies on the resolvent expansion, which is singular due to the presence of a resonance. Specializing to pure power focusing subcritical perturbations we demonstrate, via variational arguments, and for a certain range of powers, the existence of a ground state solitary wave, which is then shown to be the previously constructed solution. Finally, we pres…
In this paper, the Choquard equation with an inverse-square potential and both focusing and defocusing nonlinearities in the energy-subcritical regime is investigated. For all the cases, the local well-posedness result in $H^1(\mathbb{R}^N)$ is established. Moreover, the global existence result for arbitrary initial values is proved in the defocusing case while a global existence/blowup dichotomy below the ground state is established in the focusing case.
We prove global existence and stability of solution to the mass-critical stochastic nonlinear Schrödinger equation in $d=1$ at $L^2$ regularity. Our construction starts with the existence of solution to the truncated subcritical problem. With the presence of truncation, we construct the solution to the critical equation as the limit of subcritical solutions. We then obtain uniform bounds on the solutions to the truncated critical problems that allow us to remove truncation in the limit.
In this paper, we study the focusing and defocusing energy--subcritical, nonlinear wave equation in $\mathbb{R}^{1+d}$ with radial initial data for $d = 4,5$. We prove that if a solution remains bounded in the critical space on its interval of existence, then the solution exists globally and scatters at $\pm \infty$. The proof follows the concentration compactness/rigidity method initiated by Kenig and Merle, and the main obstacle is to show the nonexistence of nonzero solutions with a certain compactness property. A main novelty of this work is the use of a simple virial argument to rule out …
We study the nonlinear Schrödinger equation with an inverse-square potential in dimensions $3\leq d \leq 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in $H^1$ for arbitrary data.
We consider the energy-supercritical nonlinear wave equation $u_{tt}-Δu+|u|^2u=0$ with defocusing cubic nonlinearity in dimension $d=5$ with no radial assumption on the initial data. We prove that a uniform-in-time {\it a priori} bound on the critical norm implies that solutions exist globally in time and scatter at infinity in both time directions. Together with our earlier works in dimensions $d\geq 6$ with general data and dimension $d=5$ with radial data, the present work completes the study of global well-posedness and scattering in the energy-supercritical regime for the cubic nonlineari…
In this paper, we continue our study [16] on the long time dynamics of radial solutions to defocusing energy critical wave equation with a trapping radial potential in 3 + 1 dimensions. For generic radial potentials (in the topological sense) there are only finitely many steady states which might be either stable or unstable. We first observe that there can be stable excited states (i.e., a steady state which is not the ground state) if the potential is large and attractive, although all small excited states are unstable. We prove that the set of initial data for which solutions scatter to any…
We study the stochastic Nonlinear Schrödinger system with multiplicative white noise in energy space $H^1$. Based on deterministic and stochastic Strichartz estimates, we prove the local well-posedness and uniqueness of mild solution. Then we prove the global well-posedness in the mass subcritical case and the defocusing case. For the mass subcritical case, we also investigate the global existence when the $L^2$ norm of initial value is small enough. In addition, we study the blow-up phenomenon and give a sharp criteria.
Scattering for Short-Range Mass-Subcritical Choquard Equations
Mathematics and Science Department, AUBG Blagoevgrad（布拉格耶夫格拉德美国学院数学与科学系）；Hausdorff Center for Mathematics, HCM Bonn（波恩豪斯多夫数学中心）；Trieste Institute for Theoretical Quantum Technologies, TQT Trieste（的里雅斯特理论量子技术研究所）；Institute of Mathematics and Informatics, Bulgarian Academy of Science, Sofia（索菲亚保加利亚科学院数学与信息学研究所）
We generalize the pointwise decay estimates for large data solutions of the defocusing semilinear wave equations which we obtained earlier under restriction to spherical symmetry. Without the symmetry the conformal transformation we use provides only a weak decay. This can, however, in the next step be improved to the optimal decay estimate suggested by the radial case and small data results. This is the first result of that kind.
We study the asymptotic behavior of large data solutions in the energy space $H := H^1(\R^d)$ in very high dimension $d \geq 11$ to defocusing Schrödinger equations $i u_t + Δu = |u|^{p-1} u + Vu$ in $\R^d$, where $V \in C^\infty_0(\R^d)$ is a real potential (which could contain bound states), and $1+\frac{4}{d} < p < 1+\frac{4}{d-2}$ is an exponent which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show that as $t \to +\infty$, these solutions split into a radiation term that evolves according to the linear Schrödinger equation, and a remainder which con…
In this work we consider the energy subcritical 3D wave equation $\partial_t^2 u - Δu = \pm |u|^{p-1} u$ and discuss its (weakly) non-radiative solutions, i.e. the solutions defined in an exterior region $\{(x,t): |x|>|t|+R\}$ with $R\geq 0$ satisfying \[ \lim_{t\rightarrow \pm\infty} \int_{|x|>|t|+R} \left(|\nabla u(x,t)|^2 + |u_t(x,t)|^2\right) dx = 0. \] It has been known that any radial weakly non-radiative solution to the linear wave equation is a multiple of $1/|x|$. In addition, any radial weakly non-radiative solutions $u$ to the energy critical wave equation must possess a similar asy…
We consider the defocusing, $\dot{H}^1$-critical Hartree equation for the radial data in all dimensions $(n\geq 5)$. We show the global well-posedness and scattering results in the energy space. The new ingredient in this paper is that we first take advantage of the term $\displaystyle - \int_{I}\int_{|x|\leq A|I|^{1/2}}|u|^{2}Δ\Big(\frac{1}{|x|}\Big)dxdt$ in the localized Morawetz identity to rule out the possibility of energy concentration, instead of the classical Morawetz estimate dependent of the nonlinearity.
In this paper, we study the dynamics of subcritical threshold solutions for focusing energy critical NLS on $\mathbb{R}^d$ ($d\geq 5$) with nonradial data. This problem with radial assumption was studied by T. Duyckaerts and F. Merle in \cite{DM} for $d=3,4,5$ and later by D. Li and X. Zhang in \cite{LZ} for $d \geq 6$. We generalize the conclusion for the subcritical threshold solutions by removing the radial assumption for $d\geq 5$. A key step is to show exponential convergence to the ground state $W(x)$ up to symmetries if the scattering phenomenon does not occur. Remarkably, an interactio…
