Changxing Miao, Guixiang Xu, Lifeng Zhao
We establish global existence, scattering for radial solutions to the energy-critical focusing Hartree equation with energy and $\dot{H}^1$ norm less than those of the ground state in $\mathbb{R}\times \mathbb{R}^d$, $d\geq 5$.
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We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree-Fock and Hartree-Fock-Bogoliubov type equations, which describe the evolution of attractive fermionic systems (e. g. white dwarfs). Our main results are twofold: First, we extend the recent blowup result of [Hainzl and Schlein, Comm. Math. Phys. \textbf{287} (2009), 705--714] to Hartree-Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree-Fock-Bogoliubov mode…
We characterize the dynamics of the finite time blow up solutions with minimal mass for the focusing mass critical Hartree equation with $H^1(\mathbb{R}^4)$ data and $L^2(\mathbb{R}^4)$ data, where we make use of the refined Gagliardo-Nirenberg inequality of convolution type and the profile decomposition. Moreover, we also analyze the mass concentration phenomenon of such blow up solutions.
We consider $L^2$-critical focusing nonlinear Schroedinger equations with Hartree type nonlinearity $$i \pr_t u = -\DD u - \big (Φ\ast |u|^2 \big) u \quad {in $\RR^4$},$$ where $Φ(x)$ is a perturbation of the convolution kernel $|x|^{-2}$. Despite the lack of pseudo conformal invariance for this equation, we prove the existence of critical mass finite-time blowup solutions $u(t,x)$ that exhibit the pseudo-conformal blowup rate $$ \| \nabla u(t) \|_{L^2_x} \sim \frac{1}{|t|} \quad {as} \quad t \nearrow 0 . $$ Furthermore, we prove the finite-codimensional stability of this conformal blow up, by…
We study finite-time blow-up for pseudo-relativistic Hartree- and Hartree-Fock equations, which are model equations for the dynamical evolution of white dwarfs. In particular, we prove that radially symmetric initial configurations with negative energy lead to finite-time blow-up of solutions. Furthermore, we derive a mass concentration estimate for radial blow-up solutions. Both results are mathematically rigorous and are in accordance with Chandrasekhar's physical theory of white dwarfs, stating that stellar configurations beyond a certain limiting mass lead to ``gravitational collapse'' of …
We consider the gravitational collapse for neutron stars in the Hartree-Fock-Bogoliubov theory. We prove that when the number particle becomes large and the gravitational constant is small such that the attractive interaction strength approaches the Chandrasekhar limit mass slowly, the minimizers develop a universal blow-up profile. It is given by the Lane-Emden solution.
We study a class of Hartree type equations and prove a quantitative blow up rate for their blow up solutions. This is an analogue of the result by Merle and Raphaël on 3d NLS.
Negative energy blowup results for the focusing Hartree hierarchy via identities of virial and localized virial type
本研究建立了Hartree层级解的位力与局部位力恒等式，结合非线性薛定谔方程研究中的论证方法、量子de Finetti定理及截断引理，证明了无有限方差假设下特定负能量解的有限时间爆破性质。
We consider the fractional Schrödinger equations with focusing Hartree type nonlinearity. When the energy is negative, we use a Glassey's virial type argument to show the finite time blow-up of solutions.
We prove blow-up in finite time for radially symmetric solutions to the pseudo-relativistic Hartree-Fock equation with negative energy. The non-linear Hartree-Fock equation is commonly used in the physics literature to describe the dynamics of white dwarfs. We extend thereby recent results by Fröhlich and Lenzmann, who established in \cite{FL1,FL2} blow-up for solutions to the pseudo-relativistic Hartree equation. As key ingredient for handling the exchange term we use the conservation of the expectation of the square of the angular momentum operator.
We consider the following nonlinear Schrödinger equation with a Hartree nonlinearity: \[ i\frac{\partial u}{\partial t}+Δu+|u|^{\frac{4}{N}}u\pm\left(\frac{1}{|x|^{2σ}}\star|u|^2\right)u=0 \] in $\mathbb{R}^N$. We are interested in the existence and behaviour of minimal mass blow-up solutions. Previous studies have shown the existence of minimal mass blow-up solutions with an inverse power potential and investigated the behaviour of the solution. In this paper, we investigate Hartree nonlinearity, which is a nonlinear term similar to the inverse power-type potential in terms of scaling.
Using the same induction on energy argument in both frequency space and spatial space simultaneously as in \cite{CKSTT07}, \cite{RyV05} and \cite{Vi05}, we obtain global well-posedness and scattering of energy solutions of defocusing energy-critical nonlinear Hartree equation in $\mathbb{R}\times \mathbb{R}^n$($n\geq 5$), which removes the radial assumption on the data in \cite{MiXZ07a}. The new ingredients are that we use a modified long time perturbation theory to obtain the frequency localization (Proposition \ref{freqdelocaimplystbound} and Corollary \ref{frequencylocalization}) of the min…
In this paper we are interested in the following critical Hartree equation \begin{equation*} \begin{cases} -Δu =\displaystyle{\Big(\int_Ω\frac{u^{2_μ^\ast} (ξ)}{|x-ξ|^μ}dξ\Big)u^{2_μ^\ast-1}}+\varepsilon u ,~~~\text{in}~Ω,\\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}~\partialΩ, \end{cases} \end{equation*} where $N\geq4$, $0<μ\leq4$, $\varepsilon>0$ is a small parameter, $Ω$ is a bounded domain in $\mathbb{R}^N$, and $2_μ^\ast=\frac{2N-μ}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. By establishing various versions of local Pohozaev ident…
We investigate the focusing $\dot H^{1/2}$-critical nonlinear Schrödinger equation (NLS) of Hartree type $i\partial_t u + Δu = -(|\cdot|^{-3} \ast |u|^2)u$ with $\dot H^{1/2}$ radial data in dimension $d = 5$. It is proved that if the maximal life-span solution obeys $\sup_{t}\big\||\nabla|^{1/2}u\big\|_2 < \frac{\sqrt{6}}{3} \big\||\nabla|^{1/2}Q\big\|_2$, where $Q$ is the positive radial solution to the elliptic equation(\ref{e14}). Then the solution is global and scatters.
For the energy critical focusing wave equation in three dimensions we establish the existence of blowup solutions via an "exotic" dynamical rescaling of the ground state solution W(x). The scaling law is not of the pure-power type, but rather eternally oscillates between such classes. Moreover, there is a continuum of these scaling laws.
We consider the focusing nonlinear Schrödinger equation with the critical inverse square potential. We give the first proof of the uniqueness of the ground state solution. Consequently, we obtain a sharp Hardy-Gagliardo-Nirenberg interpolation inequality. Moreover, we provide a complete characterization for the minimal mass blow-up solutions to the time dependent problem.
Stable blowup for focusing semilinear wave equations in all dimensions
本文研究聚焦半线性波动方程的爆破现象，首次在所有空间维度和所有超线性幂次下，证明ODE爆破机制在无径向对称限制时的非线性渐近稳定性，覆盖整个能量超临界范围。
This paper addresses the existence of codimension one stable manifolds for the pseudo-conformal blow-up solution for critical one-dimensional NLS. By the work of Perelman and Merle, Raphael, the blow-up rate of these solutions is far from the generic one. However, Bourgain,Wang and Perelman formulated the possibility of codimension one stable manifolds for the non-generic blow-up solutions. This paper established the existence of such manifolds, albeit only in the measurable category.
On soliton clusters and collision blow up for the $L^2$-critical Hartree equation
研究$\mathbb{R}^{1+4}$中$L^2$临界Hartree方程的多孤子解，通过$m$体律近似轨迹，得到孤子簇渐近遵循双曲-抛物轨迹，并利用伪共形不变性证明任意数量孤子簇在指定点同时碰撞的有限时间爆破。
In this paper we report on numerical studies of formation of singularities for the semilinear wave equations with a focusing power nonlinearity $u_{tt} - Δu = u^{p}$ in three space dimensions. We show that for generic large initial data that lead to singularities, the spatial pattern of blowup can be described in terms of linearized perturbations about the fundamental self-similar (homogeneous in space) solution. We consider also non-generic initial data which are fine-tuned to the threshold for blowup and identify critical solutions that separate blowup from dispersal for some values of the e…
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…
Stable blowup for the focusing energy critical nonlinear wave equation under random perturbations
本文研究了三维径向聚焦能量临界非线性波动方程，通过相似坐标中的概率 Strichartz 估计，证明了能量空间以下的随机扰动下 ODE 爆破的稳定性。
We prove the existence of energy solutions of the energy critical focusing wave equation in R^3 which blow up exactly at x=t=0. They decompose into a bulk term plus radiation term. The bulk is a rescaled version of the stationary "soliton" type solution of the NLW. The construction depends crucially on the renormalization procedure of the "soliton" which we introduced in our companion paper on the wave map problem.
Multisoliton solutions and blow up for the $L^2$-critical Hartree equation
针对$L^2$-临界Hartree方程，构造了轨迹渐近服从反平方势多体定律的多孤子解，并利用伪共形对称性证明了有限时间碰撞爆破和多个点爆破。
We consider solutions of the focusing cubic and quintic Gross-Pitaevskii (GP) hierarchies. We identify an observable corresponding to the average energy per particle, and we prove that it is a conserved quantity. We prove that all solutions to the focusing GP hierarchy at the $L^2$-critical or $L^2$-supercritical level blow up in finite time if the energy per particle in the initial condition is negative. Our results do not assume any factorization of the initial data.
We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential $iu_t + Δu + (|x|^{-b} \ast |u|^p)|u|^{p-2}u=0, x \in \mathbb{R}^N$.We establish the local well-posedness at the non-conserved critical regularity $\dot{H}^{s_c}$ for $s_c \geq 0$, which also includes the energy-supercritical regime $s_c>1$ (thus, complementing the work in [3], where the authors obtained the $H^1$ well-posedness in the intercritical regime together with classification of solutions under the mass-energy threshold). We next extend the local theory to global: for sm…
In \cite{LiMZ:e-critical Har, MiaoXZ:09:e-critical radial Har}, the dynamics of the solutions for the focusing energy-critical Hartree equation have been classified when $E(u_0)<E(W)$, where $W$ is the ground state. In this paper, we continue the study on the dynamics of the radial solutions with the threshold energy. Our arguments closely follow those in \cite{DuyMerle:NLS:ThresholdSolution, DuyMerle:NLW:ThresholdSolution, DuyRouden:NLS:ThresholdSolution, LiZh:NLS, LiZh:NLW}. The new ingredient is that we show that the positive solution of the nonlocal elliptic equation in $L^{\frac{2d}{d-2}}…
We consider the fractional Hartree equation in the $L^2$-supercritical case, and we find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}<M[Q]^{\frac{s-s_c}{s_c}}E[Q]$ and $M[u_{0}]^{\frac{s-s_c}{s_c}}\|u_{0}\|^2_{\dot H^s}<M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s}$, then the solution $u(t)$ is globally well-posed and scatters; if $ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}]<M[Q]^{\frac{s-s_c}{s_c}}E[Q]$ and $M[u_{0}]^{\frac{s-s_c}{s_c}}\|u_{0}\|^2_{\dot H^s}>M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s}$, the solution $u(t)$ blow…
Classification of minimal-mass blowup solutions to the mass-critical Hartree equation at the $L^2$-critical level
该文对质量临界Hartree方程在L²临界水平下的最小质量爆破解进行分类，证明全局最小质量非散射解为孤波、有限时间最小质量爆破解为孤波的伪共形像，推广了相关结果并消除了径向假设。
Focusing $Φ^4_3$-model with a Hartree-type nonlinearity
本文研究带Hartree型非线性项的三维聚焦$Φ^4_3$模型，完成了不同$β$取值下对应吉布斯测度的构造与非归一化证明，确立了临界相变，并验证了其随机量子化方程的全局适定性与测度不变性。
We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, a Schrödinger-type equation with a nonlocal, convolution-type nonlinearity: $iu_t+Δu +\left(|x|^{-(d-2)} \ast |u|^{p} \right) |u|^{p-2}u = 0, x \in \mathbb{R}^d$. First, we consider the $L^2$-critical case in dimensions d=3, 4, 5, 6, 7 and obtain that a generic blow-up has a self-similar structure and exhibits not only the square root blowup rate but also the log-log correction (via asymptotic analysis and functional fitting). In this setting we also study blow-up profiles and show that generic blow-up s…
We study dynamics near the threshold for blowup in the focusing nonlinear Klein-Gordon equation $u_{tt}-u_{xx} + u - |u|^{2α} u =0$ on the line. Using mixed numerical and analytical methods we find that solutions starting from even initial data, fine-tuned to the threshold, are trapped by the static solution $S$ for intermediate times. The details of trapping are shown to depend on the power $α$, namely, we observe fast convergence to $S$ for $α>1$, slow convergence for $α=1$, and very slow (if any) convergence for $0<α<1$. Our findings are complementary with respect to the recent rigorous ana…
Blow-up asymptotics for a critical Hartree-type Brézis--Nirenberg problem in dimension three
研究三维临界Hartree问题，在自然非退化假设下，推导\(u_{\varepsilon}\)精确渐近展开，确定爆破率与浓度点，获浓度点附近及远离时的逐点爆破行为。
This paper studies the asymptotic behavior of global solutions to the generalized Hartree equation $$i\dot u+Δu+(I_α*|\cdot|^b|u|^p)|x|^b|u|^{p-2}u=0 .$$ Indeed, using a new approach due to \cite{dm}, one proves the scattering of the above inhomogeneous Choquard equation in the mass-super-critical and energy sub-critical regimes with radial setting.
We study the collapse of a many-body system which is used to model two-component Bose-Einstein condensates with attractive intra-species interactions and either attractive or repulsive inter-species interactions. Such a system consists a mixture of two different species for $N$ identical bosons in $\mathbb R^2$, interacting with potentials rescaled in the mean-field manner $-N^{2β-1}w^{(σ)}(N^βx)$ with $\int_{\mathbb R^{2}}w^{(σ)}(x){\rm d}x=1$. Assuming that $0<β<1/2$, we first show that the leading order of the quantum energy is captured correctly by the Gross-Pitaevskii energy. Secondly, we…
Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent
本文研究了非局部方程中稍亚临界指数单破裂解的定性性质，通过局部Pohozaev恒等式和破裂分析，提供了特征值估计并推导了单气泡解的Morse指数。
We study the blowup behavior for the focusing energy-supercritical semilinear wave equation in 3 space dimensions without symmetry assumptions on the data. We prove the stability of the ODE blowup profile.
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.
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…
We consider the nonlinear wave equation $i \partial_t u= \sqrt{-Δ+ m^2} u - (|x|^{-1} \ast |u|^2) u$ on $\RR^3$ modelling the dynamics of (pseudo-relativistic) boson stars. For spherically symmetric initial data, $u_0(x) \in C^\infty_{\mathrm{c}}(\RR^3)$, with negative energy, we prove blow-up of $u(t,x)$ in $H^{1/2}$-norm within a finite time. Physically, this phenomenon describes the onset of "gravitational collapse" of a boson star. We also study blow-up in external, spherically symmetric potentials and we consider more general Hartree-type nonlinearities. As an application, we exhibit inst…
We consider the Gross-Pitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation $Ω$. First we study the behavior of the ground state when the coupling constant approaches $a\_*$ , the critical strength of the cubic nonlinearity for the focusing nonlinear Schr{ö}dinger equation. We prove that blow-up always happens at the center of the trap, with the blow-up profile given by the Gagliardo-Nirenberg solution. In particular, the blow-up scenario is independent of $Ω$, to leading order. …
We consider the focusing mass-critical nonlinear Schrödinger equation and prove that blowup solutions to this equation with initial data in $H^s(\R^d)$, $s > s_0(d)$ and $d\geq 3$, concentrate at least the mass of the ground state at the blowup time. This extends recent work by J. Colliander, S. Raynor, C. Sulem, and J. D. Wright, \cite{crsw}, T. Hmidi and S. Keraani, \cite{hk}, and N. Tzirakis, \cite{tzirakis}, on the blowup of the two-dimensional and one-dimensional mass-critical focusing NLS below the energy space to all dimensions $d\ge 3$.
We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Hartree equation $iu_t+Δu=\pm(|x|^{-2}*|u|^2)u$ for large spherically symmetric $L^2_x(\Bbb{R}^d)$ initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state.
In this paper we prove rigidity for blowup solutions to the focusing, mass-critical nonlinear Schr{ö}dinger equation in dimensions $2 \leq d \leq 15$ with mass equal to the mass of the soliton. We prove that the only such solutions are the solitons and the pseudoconformal transformation of the solitons.
We study the global behavior of solutions to the focusing generalized Hartree equation with $H^1$ data at mass-energy threshold in the inter-range case. In the earlier works of Arora-Roudenko [Arora-Roudenko 2021], the behavior of solutions below the mass-energy threshold was classified. In this paper, we first exhibit three special solutions: $e^{it} Q$, $Q^\pm$, where $Q^\pm$ exponentially approach to the $e^{it} Q$ in the positive time direction, $Q^+$ blows up and $Q^-$ scatters in the negative time direction. Then we classify solutions at this threshold, showing that they behave exactly a…
In this paper we study dynamical properties of blowup solutions to the focusing intercritical (mass-supercritical and energy-subcritical) nonlinear fourth-order Schrödinger equation. We firstly establish the profile decomposition of bounded sequences in $\dot{H}^{γ_{\text{c}}} \cap \dot{H}^2$. We also prove a compactness lemma and a variational characterization of ground states related to the equation. As a result, we obtain the $\dot{H}^{γ_{\text{c}}}$-concentration of blowup solutions with bounded $\dot{H}^{γ_{\text{c}}}$-norm and the limiting profile of blowup solutions with critical $\dot{…
Ground states of a coupled pseudo-relativistic Hartree system: existence and concentration behavior
本文研究了带捕获势的耦合伪相对论Hartree系统的基态，通过约束最小化问题完全分类了基态存在性，并精确分析了耦合系数趋于临界值时极小子的爆破与集中行为及最优爆破速率。
Radial 3D Focusing Energy Critical INLS equations with defocusing perturbation: Ground states, Scattering, and Blow-up
本文研究径向3D下具聚焦能量临界非线性与散焦扰动的非均匀Schrödinger方程，建立了基态存在性及能量阈值下散射与爆破的二分性，揭示了竞争非线性项对解动力学的影响。
Asymptotic behavior of solutions to critical Hartree equations with isolated singularities
本文研究临界 Hartree 方程在带孔域上奇异解的局部渐近行为，通过爆破分析和积分移动球面法建立奇点附近的尖锐估计与径向对称性，推广了经典定理。
Self-similar blowup for mass supercritical Schrödinger equations
本文研究三维质量超临界聚焦非线性薛定谔方程，证明幂次接近三次时存在自相似解，并推广三次情形结论，表明自相似爆破在方程扰动下保持稳定。
We study the Cauchy problem for a system of two coupled nonlinear focusing Schroedinger equations arising in nonlinear optics. We discuss when the solutions are global in time or blow-up in finite time. Some results, in dependence of the data of the problem, are proved; in particular we give a bound, depending on the coupling parameter, for the blow-up threshold.
Scattering Criteria for the Three-Dimensional Focusing Energy-Critical Generalized Hartree Equation
针对三维聚焦能量临界广义Hartree方程，在基态阈值下建立两个非径向散射准则，通过结合局域Hartree位力恒等式与临界Hardy-Littlewood-Sobolev估计完成证明。
Scattering and Blow-Up in Both Time Directions Above the Ground State for the Focusing Nonlinear Schrödinger Equation
本文针对临界间聚焦NLS方程，构造了基态以上质量-能量任意大且不由伽利略变换产生的双时间方向散射解，还得到双时间爆破新判据与单时间散射、爆破的改进充分条件。
We consider the inhomogeneous nonlinear Schrödinger equation (INLS) in $\mathbb{R}^N$, $N \geq 1$, $$i \partial_t u + Δu + |x|^{-b} |u|^{p-1}u = 0,$$ with finite-variance initial data $u_0 \in H^1(\mathbb{R}^N)$. We extend the dichotomy between scattering and blow-up for solutions above the mass-energy threshold (and with arbitrarily large energy). We also show other two blow-up criteria, wich are valid in any mass-supercritical setting, given there is local well-posedness.
We consider the focusing $5$d Hartree equation, which is $L^2$-supercritical, with finite energy initial data, and investigate the solutions at the mass-energy threshold. We establish the existence of special solutions following the work of Duyckaerts-Roudenko [11] for the $3$d focusing cubic nonlinear Schrödinger equation (NLS). In particular, apart from the ground state solution $Q$, which is global but non-scattering, there exist special solutions $Q^+$ and $Q^-$, which in one time direction approach $Q$ exponentially, and in the other time direction $Q^+$ blows up in finite time and $Q^-$ …
We study the focusing nonlinear Schrödinger equation in the $L^2$-supercritical regime with finite energy and finite variance initial data. We investigate solutions above the energy (or mass-energy) threshold. In our first result, we extend the known scattering versus blow-up dichotomy above that threshold for finite variance solutions in the energy-subcritical and energy-critical regimes, obtaining scattering and blow-up criteria for solutions with arbitrarily large mass and energy. As a consequence, we characterize the behavior of the ground state initial data modulated by a quadratic phase.…
This paper studies the inhomogeneous fractional Schödinger equation $$i\dot u-(-Δ)^s u=\pm(I_α*|\cdot|^b|u|^p)|x|^b|u|^{p-2}u.$$ In the mass super-critical and energy sub-critical regimes, using a Gagliardo-Nirenberg adapted to the above problem, the standing waves give a sharp threshold of global existence versus finite time blow-up of solutions.
We study a class of two-dimensional non-linear Schrödinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces that are necessary for the study of ground states and evolution flow. We include in our treatment both mass sub-critical and mass critical Hartree non-linearities. Our analysis is two-fold: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy s…
We prove global well-posedness, scattering and blow-up results for energy-subcritical focusing nonlinear Schrödinger equations on the hyperbolic space. We show in particular the existence of a critical element for scattering for all energy-subcritical power nonlinearities. For mass-supercritical nonlinearity, we show a scattering vs blow-up dichotomy for radial solutions of the equation in low dimension, below natural mass and energy thresholds given by the ground states of the equation. The proofs are based on trapping by mass and energy, compactness and rigidity, and are similar to the ones …
We consider the wave equation with a focusing cubic nonlinearity in higher odd space dimensions without symmetry restrictions on the data. We prove that there exists an open set of initial data such that the corresponding solution exists in a backward light-cone and approaches the ODE blowup profile.
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…
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