We consider the Cauchy problem for the biharmonic (i.\,e.~fourth-order) NLS with focusing nonlinearity given by $i \partial_t u = Δ^2 u - μΔu -|u|^{2 σ} u$ for $(t,x) \in [0,T) \times \mathbb{R}^d$, where $0 < σ<\infty$ for $d \leq 4$ and $0 < σ\leq 4/(d-4)$ for $d \geq 5$; and $μ\in \mathbb{R}$ is some parameter to include a possible lower-order dispersion. In the mass-supercritical case $σ> 4/d$, we prove a general result on finite-time blowup for radial data in $H^2(\mathbb{R}^d)$ in any dimension $d \geq 2$. Moreover, we derive a universal upper bound for the blowup rate for suitable $4/d < σ< 4/(d-4)$. In the mass-critical case $σ=4/d$, we prove a general blowup result in finite or infinite time for radial data in $H^2(\mathbb{R}^d)$. As a key ingredient, we utilize the time evolution of a nonnegative quantity, which we call the (localized) Riesz bivariance for biharmonic NLS. This construction provides us with a suitable substitute for the variance used for classical NLS problems. In addition, we prove a radial symmetry result for ground states for the biharmonic NLS, which may be of some value for the related elliptic problem.
We prove there exist solutions to the focusing cubic nonlinear Schrödinger equation in three dimensions that blowup on a circle, in the sense of L^2 concentration on a ring, bounded H^1 norm outside any surrounding toroid, and growth of the global H^1 norm with the log-log rate. Analogous behaviour occurs in any higher dimension. That is, there exists data for which the corresponding evolution by cubic NLS explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension.
We consider fractional NLS with focusing power-type nonlinearity $$i \partial_t u = (-Δ)^s u - |u|^{2 σ} u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^N,$$ where $1/2< s < 1$ and $0 < σ< \infty$ for $s \geq N/2$ and $0 < σ\leq 2s/(N-2s)$ for $s < N/2$. We prove a general criterion for blowup of radial solutions in $\mathbb{R}^N$ with $N \geq 2$ for $L^2$-supercritical and $L^2$-critical powers $σ\geq 2s/N$. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain $Ω\subset \mathbb{R}^N$ in any dimension $N \geq 1$ and subject to exterior Dirichlet conditions. In…
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 consider the focusing $\dot H^{s_c}$-critical biharmonic Schrödinger equation, and prove a global wellposedness and scattering result for the radial data $u_0\in H^2(\mathbb R^N)$ satisfying $ M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E(Q) $ and $ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{2}\|Δu_{0}\|_{2}<\|Q\|^{\frac{2-s_c}{s_c}}_{2}\|ΔQ\|_{2}, $ where $s_c\in(0,2)$ and $Q$ is the ground state of $Δ^2Q+(2-s_c)Q-|Q|^{p-1}Q=0$.
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 consider the inhomogeneous biharmonic nonlinear Schrödinger equation $$ i u_t +Δ^2 u+λ|x|^{-b}|u|^αu = 0, $$ where $λ=\pm 1$ and $α$, $b>0$. In the subctritical case, we improve the global well-posedness result obtained in \cite{GUZPAS} for dimensions $N=5,6,7$ in the Sobolev space $H^2(\mathbb{R}^N)$. The fundamental tools to establish our results are the standard Strichartz estimates related to the linear problem and the Hardy-Littlewood inequality. Results concerning the energy-critical case, that is, $α=\frac{8-2b}{N-4}$ are also reported. More precisely, we show well-posedness and a st…
We consider the focusing inhomogeneous biharmonic nonlinear Schrödinger equation in $H^2(\mathbb{R}^N)$, \begin{equation} iu_t + Δ^2 u - |x|^{-b}|u|^αu=0 \end{equation} when $b > 0$ and $N \geq 5$. We first obtain a small data global result in $H^2$, which, in the five-dimensional case, improves a previous result from Pastor and the second author. In the sequel, we show the main result, scattering below the mass-energy threshold in the intercritical case, that is, $\frac{8-2b}{N} < α<\frac{8-2b}{N-4}$, without assuming radiality of the initial data. The proof combines the decay of the nonlinea…
We study $O(d)$-equivariant biharmonic maps in the critical dimension. A major consequence of our study concerns the corresponding heat flow. More precisely, we prove that blowup occurs in the biharmonic map heat flow from $B^4(0, 1)$ into $S^4$. To our knowledge, this was the first example of blowup for the biharmonic map heat flow. Such results have been hard to prove, due to the inapplicability of the maximum principle in the biharmonic case. Furthermore, we classify the possible $O(4)$-equivariant biharmonic maps from $\mathbf{R}^4$ into $S^4$, and we show that there exists, in contrast to…
In this paper, we study the $H^s$-stability of the log-log blowup regime (which has been completely described in a series of recent works by Merle and Raphael) for the focusing mass-critical nonlinear Schrödinger equations $i\partial_tu+Δu+|u|^\frac4du=0$ in $\mathbb{R}^d$ with $d\geq3$. We aim to extend the result in [Colliander and Raphael, Rough blowup solutions to the $L^2$ critical NLS, Math. Anna., 345(2009), 307-366.] for dimension two to the higher dimensions cases $d\geq3$, where we use the bootstrap argument in the above paper and the commutator estimates in [M. Visan and X. Zhang, O…
In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $λ\in \mathbb R$, $d\in \mathbb N$, $0<s<\min \{2+\frac{d}{2},\frac{3}{2}d\}$ and $0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is globally well-posed in $H^{s}(\mathbb R^{d})$ if $\frac{8-2b}{d}<σ< σ_{c}(s)$ and the initial data is sufficiently small, where $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and…
We consider singular solutions of the biharmonic NLS. In the L^2-critical case, the blowup rate is bounded by a quartic-root power law, the solution approaches a self-similar profile, and a finite amount of L^2-norm, which is no less than the critical power, concentrates into the singularity ("strong collapse"). In the L^2-critical and supercritical cases, we use asymptotic analysis and numerical simulations to characterize singular solutions with a peak-type self-similar collapsing core. In the critical case, the blowup rate is slightly faster than a quartic-root, and the self-similar profile…
Let $d\geq 4$ and let $u$ be a global solution to the focusing mass-critical nonlinear Schrödinger equation $iu_t+Δu=-|u|^{\frac 4d}u$ with spherically symmetric $H_x^1$ initial data and mass equal to that of the ground state $Q$. We prove that if $u$ does not scatter then, up to phase rotation and scaling, $u$ is the solitary wave $e^{it}Q$. Combining this result with that of Merle \cite{merle2}, we obtain that in dimensions $d\geq 4$, the only spherically symmetric minimal-mass blowup solutions are, up to phase rotation and scaling, the pseudo-conformal ground state and the solitary wave.
We consider the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation in $\mathbb{R}^N$, $$i \partial_t u +Δ^2 u -|x|^{-b} |u|^{2σ}u = 0,$$ where $σ>0$ and $b>0$. We first study the local well-posedness in $\dot H^{s_c}\cap \dot H^2 $, for $N\geq 5$ and $0<s_c<2$, where $s_c=\frac{N}{2}-\frac{4-b}{2σ}$. Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in $\dot H^{s_c}\cap \dot H^2$ with $0\leq s_c<2$. Finally, we study the phenomenon of $L^{σ_c}$-norm concentration for finite time blow up solutions w…
We consider ground states solutions $u \in H^2(\mathbb{R}^N)$ of biharmonic (fourth-order) nonlinear Schrödinger equations of the form $$ Δ^2 u + 2a Δu + b u - |u|^{p-2} u = 0 \quad \mbox{in $\mathbb{R}^N$} $$ with positive constants $a, b > 0$ and exponents $2 < p < 2^*$, where $2^* = \frac{2N}{N-4}$ if $N > 4$ and $2^* = \infty$ if $N \leq 4$. By exploiting a connection to the adjoint Stein--Tomas inequality on the unit sphere and by using trial functions due to Knapp, we prove a general symmetry breaking result by showing that all ground states $u\in H^2(\mathbb{R}^N)$ in dimension $N \geq …
We consider the minimal mass $m_0$ required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation $iu_t + Δu = μ|u|^{4/d} u$ to blow up. If $m_0$ is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in $L^2_x(\R^d)$ is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, \cite{keraani}, in dimensions 1, 2 and Begout and Vargas, \cite{begout}, in di…
We study the focusing mass-subcritical biharmonic nonlinear Schrödinger equation (BNLS) on the product space $\mathbb{R}_x^d\times\mathbb{T}_y^n$. Following the crucial scaling arguments introduced in \cite{TTVproduct2014} we establish existence and stability results for the normalized ground states of BNLS. Moreover, in the case where lower order dispersion is absent, we prove the existence of a critical mass number $c_0\in(0,\infty)$ that sharply determines the $y$-dependence of the deduced ground states. In the mixed dispersion case, we encounter a major challenge as the BNLS is no longer s…
Consider a branch of unstable solitons of NLS whose linearized operators have one pair of simple real eigenvalues in addition to the zero eigenvalue. Under radial symmetry and standard assumptions, solutions to initial data from a neighbourhood of the branch either converge to a soliton, or exit a larger neighbourhood of the branch transversally. The qualitative dynamic near a branch of unstable solitons is irrespective of whether blowup eventually occurs, which has practical implications for the description of blowup of NLS with supercritical nonlinearity.
We consider the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,\;u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $λ\in \mathbb R$, $d\in \mathbb N$, $0\le s<\min\left\{2+\frac{d}{2},d\right\}$, $0<b<\min \left\{4,\; d-s,\; 2+\frac{d}{2}-s \right\}$ and $0<σ\le σ_{c}(s)$ with $σ<\infty$. Here $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $σ_{c}(s)=\infty$ if $s\ge \frac{d}{2}$. First, we give some remarks on Sobolev-Lorentz spaces and extend the chain rule under Lorentz norms for the fractional Laplacian $(-Δ)^{s/2}$…
We prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schrödinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the gain and dissipation are equal. We address two cases: in the first model all nonlinear coefficients (i.e. the ones describing self-action and non-linear coupling) correspond to attractive (focusing) nonlinearities, and in the second case the NLS equation with gain has attractive nonlinearity while the NLS equation with dissipation has repulsive (defocusing) nonli…
Rough log-log blowup solutions to mass-critical NLS in higher dimensions $d\geq 3$
Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所）；School of Mathematics and Physics, University of Science and Technology Beijing（北京科技大学数学与物理学院）
We consider the cubic and quintic nonlinear Schrödinger equations (NLS) under the $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for solutions to NLS at critical regularity for all dimensions. Thus, together with [18,19], the unconditional uniqueness problems for $H^{1}$-critical and $H^{1}$-supercritical cubic and quintic NLS are completely and uniformly resolved at critical regularity for these domains. One application of our theorem is to prove that defocusing blowup solutions of the type in [54…
We mainly consider the focusing biharmonic Schrödinger equation with a large radial repulsive potential $V(x)$: \begin{equation*} \left\{ \begin{aligned} iu_{t}+(Δ^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {\bf{R}\times{\bf{R}}^{N}}, u(0, x)=u_{0}(x)\in H^{2}({\bf{R}}^{N}), \end{aligned}\right. \end{equation*} If $N>8$, \ $1+\frac{8}{N}<p<1+\frac{8}{N-4}$ (i.e. the $L^{2}$-supercritical and $\dot{H}^{2}$-subcritical case ), and $\langle x\rangle^β\big(|V(x)|+|\nabla V(x)|\big)\in L^\infty$ for some $β>N+4$, then we firstly prove a global well-posedness and scattering result for the radial data $u_0\in …
We consider the Cauchy problem of nonlinear Schrödinger equations (NLS) with almost periodic functions as initial data. We first prove that, given a frequency set $\pmbω =\{ω_j\}_{j = 1}^\infty$, NLS is local well-posed in the algebra $\mathcal{A}_{\pmbω}(\mathbb R)$ of almost periodic functions with absolutely convergent Fourier series. Then, we prove a finite time blowup result for NLS with a nonlinearity $|u|^p$, $p \in 2\mathbb{N}$. This elementary argument presents the first instance of finite time blowup solutions to NLS with generic almost periodic initial data.
We consider the blowup rate for blowup solutions to $L^2$-critical, focusing NLS with a harmonic potential and a rotation term. Under a suitable spectral condition we prove that there holds the "$\log$-$\log$ law" when the initial data is slightly above the ground state. We also construct minimal mass blowup solutions near the ground state level with distinct blowup rates.
Threshold dynamics for the 4$d$ mass-energy double critical NLS
Universidad del Valle（巴拿马谷大学）；The Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）；University of Oregon（俄勒冈大学）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
We study the global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schrödinger equation (NLS), $i \partial_t u+Δu+ |u|^{p-1}u=0, $ with initial data $u_0\in H^1,\; x \in R^n$. The nonlinearity power $p$ and the dimension $d$ are such that the scaling index $s=\frac{d}2-\frac2{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$ For solutions with $\ME[u_0]<1$ ($\ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is given. Na…
We consider the following nonlinear Schrödinger equation with the double $L^2$-critical nonlinearities \begin{align*} iu_t+Δu+|u|^\frac{4}{3}u+μ\left(|x|^{-2}*|u|^2\right)u=0\ \ \ \text{in $\mathbb{R}^3$,} \end{align*} where $μ>0$ is small enough. Our first goal is to prove the existence and the non-degeneracy of the ground state $Q_μ$. In particular, we develop an appropriate perturbation approach to prove the radial non-degeneracy property and then obtain the general non-degeneracy of the ground state $Q_μ$. We then show the existence of finite time blowup solution with minimal mass $\|u_0\|…
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.
This paper is concerned with the global well-posedness and finite time blowup problem for the 3D focusing energy-critical inhomogeneous NLS. In the previous results \cite{chkl2, chkl3} the authors considered the same problems with the spatial inhomogeneity coefficient $g$ such that $g(x) \sim |x|^{-b}$ for $0 \le b < \frac43$. Here we extend the inhomogeneous index $b$ up to $\frac32$. For this purpose, we improve the local theory and develop a new profile decomposition based on weighted space.
We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $\mathbb{R}^d$, $d=5,6$, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (i) we prove almost sure global well-posedness of the (standard) energy-critical NLS on $\mathbb{R}^d$, $d = 5, 6$, in the defocusing case, and (ii) …
In this paper we consider inhomogeneous cubic-quintic NLS in space dimension $d = 3$: $$ iu_t = -Δu + K_1(x)|u|^2u + K_2(x)|u|^4u. $$ We study local well-posedness, finite time blowup, and small data scattering and non-scattering for the ICQNLS when $K_1, K_2 \in C^4(\mathbb R^3 \setminus \{0\})$ satisfy growth condition $|\partial^j K_i(x)| \lesssim |x|^{b_i-j}\, (j = 0, 1, 2, 3, 4)$ for some $b_i \ge 0$ and for $x \neq 0$. To this end we use the Sobolev inequality for the functions $f \in H^n \,(n = 1, 2)$ such that $\||\mathbf L|^\ell f\|_{H^n} < \infty \,(\ell = 1, 2)$, where $\mathbf L$ i…
We consider the focussing energy-critical inhomogeneous nonlinear Schrödinger equation: $$ iu_t + Δu + g|u|^2u = 0, u(0)= φ\in \dot{H}^1,\;\; 0 \le g_i \le |x|g \le g_s.$$ On the road map of Kenig-Merle \cite{km} we show the global well-posedness and scattering of radial solutions under energy condition $$E_g(φ) < E_g(Q),\;\;\mbox{and}\;\; g_s\|φ\|_{\dot{H}^1}^2 < \|Q\|_{\dot{H}^1}^2,$$ where $Q$ is the solution of $ΔQ + |x|^{-1}Q^3 = 0$, together with scaling condition $|g(x)| + |x||\nabla g(x)| \lesssim |x|^{-1}$, variational condition $g_s(2-g_i) \le 1$, and rigidity condition $-g(x) \le x\…
We consider the global regularity problem for defocusing nonlinear Schrödinger systems $$ i \partial_t + Δu = (\nabla_{{\bf R}^m} F)(u) + G $$ on Galilean spacetime ${\bf R} \times {\bf R}^d$, where the field $u\colon {\bf R}^{1+d} \to {\bf C}^m$ is vector-valued, $F\colon {\bf C}^m \to {\bf R}$ is a smooth potential which is positive, phase-rotation-invariant, and homogeneous of order $p+1$ outside of the unit ball for some exponent $p >1$, and $G: {\bf R} \times {\bf R}^d \to {\bf C}^m$ is a smooth, compactly supported forcing term. This generalises the scalar defocusing nonlinear Schrödinge…
We consider the focusing fractional nonlinear Schrödinger equation \[ i\partial_t u - (-Δ)^s u = -|u|^αu, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $s \in (1/2,1)$ and $α>0$. By using localized virial estimates, we establish general blow-up criteria for non-radial solutions to the equation. As consequences, we obtain blow-up criteria in both $L^2$-critical and $L^2$-supercritical cases which extend the results of Boulenger-Himmelsbach-Lenzmann [{\it Blowup for fractional NLS}, J. Funct. Anal. 271 (2016), 2569--2603] for non-radial initial data.
