\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb, graphicx} \usepackage[abs]{overpic} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 211, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/211\hfil Cross-constrained problems] {Cross-constrained problems for nonlinear Schr\"odinger equation with harmonic potential} \author[R. Xu, C. Xu \hfil EJDE-2012/211\hfilneg] {Runzhang Xu, Chuang Xu} % in alphabetical order \address{Runzhang Xu \newline Department of Applied Mathematics, Harbin Engineering University, 150001, China} \email{xurunzh@yahoo.com.cn} \address{Chuang Xu \newline Department of Mathematiccal and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Alberta, Canada \newline Department of Mathematics, Harbin Institute of Technology, 150001, China} \email{xuchuang6305@163.com} \thanks{Submitted August 3, 2012. Published November 27, 2012.} \subjclass[2000]{78A60, 35Q55} \keywords{Cross-constrained problem; blow up; global existence; \hfill\break\indent invariant manifold; harmonic potential} \begin{abstract} This article studies a nonlinear Sch\"odinger equation with harmonic potential by constructing different cross-constrained problems. By comparing the different cross-constrained problems, we derive different sharp criterion and different invariant manifolds that separate the global solutions and blowup solutions. Moreover, we conclude that some manifolds are empty due to the essence of the cross-constrained problems. Besides, we compare the three cross-constrained problems and the three depths of the potential wells. In this way, we explain the gaps in [J. Shu and J. Zhang, Nonlinear Shr\"odinger equation with harmonic potential, Journal of Mathematical Physics, 47, 063503 (2006)], which was pointed out in [R. Xu and Y. Liu, Remarks on nonlinear Schr\"odinger equation with harmonic potential, Journal of Mathematical Physics, 49, 043512 (2008)]. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction}\label{sec1} In this paper, we study the following initial-value problem for the nonlinear Sch\"odinger equation with harmonic potential: $$\label{2} \begin{gathered} i\varphi_t+\Delta \varphi-|x|^2\varphi+|\varphi|^{p-1}\varphi=0,\quad t>0,\; x\in\mathbb{R}^N, \\ \varphi(0,x)=\varphi_0(x). \end{gathered}$$ Hereafter we will use the following notation: $\varphi(x,t):\mathbb{R}^N\times[0,T_a)\to \mathbb{C}$ is a complex valued wavefunction; $00,\; x\in\mathbb{R}^N, \\ \varphi(0,x)=\varphi_0(x). \end{gathered} It is well-known that $$\label{1} \begin{gathered} i\varphi_t+\Delta \varphi+|\varphi|^{p-1}\varphi=0,\quad t>0,\; x\in\mathbb{R}^N, \\ \varphi(0,x)=\varphi_0(x), \end{gathered}$$ is one of the basic evolution models for nonlinear waves in various branches of physics. Many papers have studied equation \eqref{1}. In \cite{JG}, Ginibre and Velo established the local existence of the Cauchy problems in the energy space$H^1(\mathbb{R}^N)$. Glassey \cite{RT}, Tsutsumi \cite{MT}, Ogawa and Tsutsumi \cite{TY,TY2} proved that for some initial data, especially for a class of sufficiently large data, the solutions of the Cauchy problem for \eqref{1} blow up in finite time. Strauss and Cazenave also mentioned this topic in their monographs \cite{W} and \cite{T} respectively. There are also many mathematicians who addressed these problems with harmonic potential. It is found that for sufficiently small initial data, the solutions of the Cauchy problem for \eqref{1} globally exist (cf. \cite{JG2,NY,CGL,JT,NKM}, etc). Zhang \cite{J1} studied the global existence of \eqref{1} and the relationship between the Schr\"odinger equation and its ground state. For \eqref{3}, Fujiwara \cite{F} proved the smoothness of Schr\"odinger kernel for potentials of quadratic growth. It is shown that quadratic potentials are the highest order potentials for local well-posedness of the equation \cite{O}. Yajima \cite{Y} showed that for super-quadratic potentials, the Schr\"odinger kernel is nowhere$C^1$. When$p>1+4/N$, Cazenave \cite{T}, Tsurumi and Wadati \cite{TT3} and Carles \cite{RC1, RC2} showed that the solutions of the Cauchy problem of \eqref{2} blow up in finite time for some initial data, especially for a class of sufficiently large initial data; while the solutions of the Cauchy problem of \eqref{2} globally exist for other initial data, especially for a class of sufficiently small initial data, see \cite{RC1}, \cite{RC2} and \cite{TT3}. When$10$for$i=1,2,3$. \end{lemma} \begin{proof} (i) For any$\varphi\in M_1\cup M_2$, we have $$\int|\nabla\varphi|^2+|\varphi|^2 dx\leqslant\frac{N(p-1)}{2(p+1)}\int|\varphi|^{p+1}dx.$$ By Sobolev embedding inequality, this implies $$\int|\nabla\varphi|^2+|\varphi|^2 dx\geqslant C.$$ Note by assumption \eqref{eH}, $$\frac{1}{2}-\frac{1}{p+1}\cdot\frac{2(p+1)}{N(p-1)}>0.$$ Hence$P(\varphi)\geqslant C>0$, which verifies$d_i>0$($i=1,2$). (ii) For$\varphi\in M_3$, we have $$\|\nabla\varphi\|_2^2\leqslant\frac{N(p-1)}{2(p+1)}\int |\varphi|^pdx,$$ which implies from Gagliardo-Nirenberg inequality and Cauchy-Schwartz inequality that there exists a constant$C(p,N)>0such that \begin{align*} C(p,N)&\leqslant \|\nabla\varphi\|_2^{\frac{Np-(N+4)}{2}} \cdot\|\varphi\|_2^{\frac{(N+2)-(N-2)p}{2}}\\ &\leqslant \frac{1}{2}\Big(\|\nabla\varphi\|_2^{Np-(N+4)} +\|\varphi\|_2^{(N+2)-(N-2)p}\Big). \end{align*} This yields\|\nabla\varphi\|_2\geqslant C>0$or$\|\varphi\|_2\geqslant C>0$. Thus $$P(\varphi)=\frac{1}{2}\|\varphi\|_2^2+\frac{Np-(N+4)}{2N(p-1)} \Big[\|\nabla\varphi\|_2+\int|x|^2|\varphi|^2dx\Big]\geqslant C>0,$$ which proves$d_3>0$. \end{proof} Next we give the invariance of some manifolds. \begin{theorem}\label{th1} For$i=1,2,3$, define \label{29} \mathcal{G}_i :=\{\psi\in H: P(\psi)0\}\cup\{0\} Then$\mathcal{G}_i$is an invariant manifold of \eqref{2}; that is, if$\varphi_0\in \mathcal{G}_i$, then the solution$\varphi(x, t)$of the Cauchy problem \eqref{2} also satisfies$\varphi(x, t)\in \mathcal{G}_i$for any$t\in [0, T)$. \end{theorem} \begin{proof} If$\varphi_0=0$, from the mass conservation law; i.e., \eqref{11}, we can find that$\varphi=0$for$t\in[0, T)$; i.e.,$\varphi(x, t)\in \mathcal {G}_i$. If$\varphi_0\neq 0$, we have$\varphi_0\in \mathcal {G}_i\backslash\{0\}$; i.e.,$P(\varphi_0)0$. By Lemma \ref{le1}, there exists a unique$\varphi(x, t)\in C([0, T); H)$with$00$for$t\in[0, T)$. Note that$I_i(\varphi_0)>0$. Arguing by contradiction, by the continuity of$I_i(\varphi)$, suppose that there were a$t_2\in[0, T)$such that$I_i(\varphi(x ,t_2 ))=0$. If$\varphi(x,t_2)=0$, then by \eqref{11}, we have$0=\int|\varphi(x,t_2)|^2 dx=\int|\varphi_0|^2 dx$, which indicates$\varphi_0=0$. Contradiction. So$\varphi(x,t_2)\neq0$, by the definition of$d_i$, we have$P(\varphi(x,t_2))\geqslant d_i$, which contradicts \eqref{1-12}. Therefore$I_i(\varphi)>0$for all$t\in[0, T)$. Combining all of the analysis above, we arrive at$\varphi(x,t)\in \mathcal {G}_i$for any$t\in [0, T)$. The proof is complete. \end{proof} By a similar argument, we can obtain the following result. \begin{theorem}\label{th2} For$i=1,2,3$, define \begin{equation*}%\label{32} \mathcal{B}_i :=\{\psi\in H: P(\psi)P(\varphi_i),\ I_i(\varphi_i)>0\ (i=1,2,3). $$For i=1,2,3, it always follows from I_i(\varphi_i)>0 that$$ \frac{1}{p+1}|\varphi_i|^{p+1} dx<\frac{2}{N(p-1)}\int |\nabla \varphi_i|^2+|\varphi_i|^2+|x|^2|\varphi_i|^2 dx $$Thus we obtain $$\label{33} \begin{split} d_i&>P(\varphi_i)\\ &=\int \frac{1}{2}|\nabla \varphi_i|^2+\frac{1}{2}|\varphi_i|^2+\frac{1}{2}|x|^2|\varphi_i|^2 -\frac{1}{p+1}|\varphi_i|^{p+1} dx\\ &> \Big(\frac{1}{2}-\frac{2}{N(p-1)}\Big)\int |\nabla \varphi_i|^2+|\varphi_i|^2+|x|^2|\varphi_i|^2 dx, \end{split}$$ which yields$$ \int |\nabla \varphi_i|^2+|\varphi_i|^2+|x|^2|\varphi_i|^2 dx<\frac{2N(p-1)d_i}{N(p-1)-4}. $$Therefore, it follows from Lemma \ref{le1} that \varphi globally exists on t\in [0, \infty). At this point, we proved this theorem. \end{proof} \begin{theorem}\label{th3} If \varphi_0\in \mathcal {B}_i\ (i=1,2,3), then the solution \varphi(x, t) of the Cauchy problem \eqref{1} blows up in finite time. \end{theorem} \begin{proof} We prove this theorem case by case. \noindent\textbf{Case I}: \varphi_0\in \mathcal {B}_1\cup\mathcal {B}_2. In this case, Theorem \ref{th1} implies that the solution \varphi(x, t) of the Cauchy problem \eqref{1} satisfies that \varphi(x, t)\in \mathcal {B}_1\cup\mathcal {B}_2 for t\in [0, T). For J(t)=\int |x|^2|\varphi|^2 dx, the definitions of P(\varphi) and I_i(\varphi) (i=1,2) imply that $$\label{39} J''(t)<-8\int|\varphi|^2 dx.$$ Then$$ J'(t)0$. Therefore there exists a $T_1\in (0,\infty)$ such that $J(t)>0$ for $t\in[0,T_1)$ and $J(T_1)=0$. By the inequality (see \cite{YJ}) $$\|\varphi\|^2\leqslant\frac{2}{N}\|\nabla\varphi\|\cdot\|x\varphi\|$$ we obtain $\lim_{t\to T_1}\|\nabla\varphi\|=\infty$, which indicates $$\underset{t\to T_1}{\lim}\|\varphi\|_H=\infty.$$ \noindent\textbf{Case ii}: $\varphi_0\in \mathcal {B}_3$. In this case, Theorem \ref{th1} implies that the solution $\varphi(x, t)$ of the Cauchy problem \eqref{1} satisfies that $\varphi(x, t)\in \mathcal {B}_3$ for $t\in [0, T)$. For $J(t)=\int |x|^2|\varphi|^2 dx$, \eqref{7} and \eqref{10} imply that $$\label{40} J''(t)<-16\int|x|^2|\varphi|^2 dx.$$ Now we show that there exists a $T_1\in (0,\infty)$ such that $J(t)>0$ for $t\in[0,T_1)$ and $J(T_1)=0$. Arguing by contradiction, suppose $\forall t\in[0,\infty)$, $J(t)>0$. Set $$g(t)=\frac{J'(t)}{J(t)}.$$ It is easy to show that $$\label{21} g'(t)=\frac{J''(t)}{J(t)}-\Big(\frac{J'(t)}{J(t)}\Big)^2<-16-g^2(t).$$ Next we like to show $g(t)\neq0$ for any $t\in[0,\infty)$. Arguing by contradiction again, suppose there is a $t_0$ such that $g(t_0)=0$. By \eqref{21}, we have $g(t)<0$ for $t\in(t_0,\infty)$. For any fixed $t_1>t_0$, dividing \eqref{21} by $g^2(t)$, we have $$\frac{g'(t)}{g^2(t)}<-\frac{16}{g^2(t)}-1<-1.$$ Further we derive $$\int_{t_1}^{t}\frac{g'(\tau)}{g^2(\tau)} d\tau<\int_{t_1}^{t}-1 d\tau,$$ namely, $$\label{111} \frac{1}{g(t)}>\frac{1}{g(t_1)}+(t-t_1),$$ which indicates that there exists a $t_2>t_1$ such that $$\label{41} g(t)>0 \quad \text{for any } t\in(t_2,\infty)$$ This contradicts $g(t)<0$ for $t\in(t_0,\infty)$. Hence we have $g(t)\neq0$ for any $t\in[0,\infty)$. By \eqref{111}, for $t\in(0,\infty)$ , we have $$\frac{1}{g(t)}>\frac{1}{g(0)}+t.$$ Hence, $J'(t)>0$ for $t\in(|\frac{1}{g(0)}|,\infty)$. Therefore $J(t)$ is increasing in $(|\frac{1}{g(0)}|,\infty)$. Let $t_0=|\frac{1}{g(0)}|$. By $$J''(t)<-16J(t)<0,$$ we have for $t>t_0$, $$J'(t)0 for t\in[0,T_1) and J(T_1)=0. Again by the inequality (see \cite{YJ})$$ \|\varphi\|^2\leqslant\frac{2}{N}\|\nabla\varphi\|\cdot\|x\varphi\| $$we obtain$$\lim_{t\to T_1} \|\nabla\varphi\|=\infty. $$So far we have shown that for the initial data \varphi\in\mathcal {B}_i , the solution of the Cauchy problem \eqref{2} blows up in finite time. This completes the proof of the theorem. \end{proof} \begin{remark} \rm It is clear that$$ \{\psi\in H, P(\psi)0$. Otherwise, there is a$0<\mu\leqslant1$such that$I_i(\mu\varphi_0)=0$. Thus$P(\mu\varphi_0)\geqslant d_i$. On the other hand, $$\|\mu\varphi_0\|^2_H=\mu^2\|\varphi_0\|^2_H<2\mu^2d_i<2d_i.$$ It follows that$P(\mu\varphi_0)0, I_j(\psi)>0 \}\cup\{0\}, \\ \label{49} \mathcal {V}_{+i,-j}: =\{\psi\in H: P(\psi)0, I_j(\psi)<0 \}, \\ \label{50} \mathcal {V}_{-i,+j}: =\{\psi\in H: P(\psi)0 \}, \\ \label{51} \mathcal {B}_{i,j}: =\{\psi\in H: P(\psi)0\},\\ \mathcal {B}_{ij}: =\{\psi\in H: P(\psi)I_i$($i=2,3$) for$\varphi\neq0$. Now we like to analyze why$\mathcal {V}_{+2,-3}=\emptyset$,$\mathcal {V}_{-2,+3}=\emptyset$,$\mathcal {V}_{+1,-i}=\emptyset$for$i=2,3$. In fact, we know that$\mathcal {V}_{+1,-2}$is a subset of both$\mathcal {G}_1$and$\mathcal {B}_2$. We have proved that$\mathcal{G}_1$is a manifold of all the global solutions of the Cauchy problem \eqref{2} for$P(\varphi)d_2>d_3$; The intersections of the three spheres and red surface represent the three manifolds$I_i(\varphi)=0$($i=1,2,3$), respectively.} \label{fig1} \end{figure} \begin{figure}[ht] \begin{center} \begin{overpic}[scale=0.42,tics=20]{fig2} \put(110,106){\makebox(0,0)[cc]{\small$I_3(\varphi)=0$}} \put(88.62,120){\makebox(0,0)[cc]{\small$I_2(\varphi)=0$}} \put(48.72,106){\makebox(0,0)[cc]{\small$I_1(\varphi)=0$}} \put(145.74,110){\makebox(0,0)[cc]{\small$O$}} \put(140.5,94.92){\makebox(0,0)[cc]{\small$I_3(\varphi)<0$}} \put(140.28,67.2){\makebox(0,0)[cc]{\small$I_2(\varphi)<0$}} \put(140.98,32.34){\makebox(0,0)[cc]{\small$I_1(\varphi)<0$}} \put(140,120.26){\makebox(0,0)[cc]{\small$I_3(\varphi)>0$}} \put(140,148.26){\makebox(0,0)[cc]{\small$I_2(\varphi)>0$}} \put(140,184.8){\makebox(0,0)[cc]{\small$I_1(\varphi)>0$}} \end{overpic} \end{center} \caption{Cross section for Figure 1} \label{fig2} \end{figure} \subsection*{Acknowledgements} This work is supported by grants: 11101102, 11101104, 11031002 from the National Natural Science Foundation of China; 20102304120022 from the Ph. D. Programs Foundation of Ministry of Education of China; 1252G020 from the Support Plan for the Young College Academic Backbone of Heilongjiang Province; A201014 from the Natural Science Foundation of Heilongjiang Province; 12521401 from the Science and Technology Research Project of Department of Education of Heilongjiang Province; and from Foundational Science Foundation of Harbin Engineering University and Fundamental Research Funds for the Central Universities. The second author was supported by China Scholarship Council and University of Alberta Doctoral Recruitment Scholarship. The authors want to thank the anonymous referee for his/her valuable suggestions which greatly improve the presentation of this article. The authors also thank Yufeng Wang and Dr. Yanyou Chai for their kind help. \begin{thebibliography}{10} \bibitem{GJ} G. Chen, J. Zhang; \emph{Remarks on global existence for the supercritical nonlinear Schr\"odinger equation with a harmonic potential}, Journal of Mathematical Analysis and Applications, 320 (2006), pp.~591--598. \bibitem{RC1} R. Carles; \emph{Critical nonlinear Schr\"odinger equation with and without harmonic potential}, Mathematical Models and Methods in Applied Sciences, 12 (2002), pp.~1513--1523. \bibitem{RC2} R. Carles; \emph{Remarks on the nonlinear Schr\"odinger equation with harmonic potential}, Annales Henri Poincar\'{e}, 3 (2002), pp.~757--772. \bibitem{T} T. Cazenave; \emph{An introduction to nonlinear Schr\"odinger equations}, Textos de Metodos Matematicos, Vol. 22, Rio de Janeiro, 1989. \bibitem {TC} T. Cazenave; \emph{Semilinear Schr\"odinger Equations}, Courant Lecture Notes in Mathematics, American Mathematical Society, Providence, Rhode Island, 2003. \bibitem{F} D. Fujiwara; \emph{Remarks on convergence of the Feynman path integrals}, Duke Mathematical Journal, 47 (1980), pp.~559--600. \bibitem{JT} J. Ginibre, T. Ozawa; \emph{Long range scattering for nonlinear Schr\"odinger and Hartree equations in space dimensions$n\geqslant2$}, Communications in Mathematical Physic, 151 (1993), pp.~619--645. \bibitem {JG} J. Ginibre, G. Velo; \emph{On a class of nonlinear Schr\"odinger equations}, Journal of Functional Analysis, 32 (1979), pp.~1--71. \bibitem{JG2} J. Ginibre, G. Velo; \emph{The global Cauchy problem for the nonlinear Schr\"odinger equation, revisited}, Annales de l'Institut Henri Poincar\'{e} Non Linear Analysis, 2 (1985), pp.~309--327. \bibitem{RT} R. T. Glassey; \emph{On the blowing up of solutions to the Cauchy problem for nonlinear Schr\"odinger equations}, Journal of Mathematical Phisics, 18 (1977), pp.~1794--1797. \bibitem{NKM} N. Hayashi, K. Nakamitsu, M. Tsutsumi; \emph{On solutions of the initial value problem for the nonlinear Schr\"odinger equations}, Journal of Functional Analysis, 71 (1987), pp.~218--245. \bibitem{NY} N. Hayashi, Y. Tsutsumi; \emph{Scattering theory for Hartree type equations}, Annales de I'institut Henri Poincar\'{e}, Physique Th\'{e}orique, 46 (1987), pp.~187--213. \bibitem{CGL} C. Kenig, G. Ponce, L. Vega; \emph{Small solution to nonlinear Schr\"odinger equations}, Annales de l'Institut Henri Poincar\'{e} Nonlinear Analysis, 10 (1993), pp.~255--288. \bibitem {TK} T. Kato; \emph{On nonlinear Shr\"odinger equations}, Annales de I'institut Henri Poincar\'{e}, Physique Th\'{e}orique 49 (1987), pp.~113--129. \bibitem{TY} T. Ogawa, Y. Tsutsumi; \emph{Blow-up of$H^1$solution for the nonlinear Schr\"odinger equation}, Journal of Differential Equations 92 (2) (1991), pp.~317--330. \bibitem{TY2} T. Ogawa, Y. Tsutsumi; \emph{Blow-up of$H^1\$ solution for the nonlinear Schr\"odinger equation with critical power nonlinearity}, Proceedings of the American Mathematical Society, 111 (2) (1991), pp.~487--496. \bibitem{O} Y. G. Oh; \emph{Cauchy problem and Ehrenfest's law of nonlinear Schr\"oinger equations with potentials}, Journal of Differential Equations, 81 (1989), pp.~255--274. \bibitem {L} L. E. Payne, D. H. Sattinger; \emph{Saddle points and instability of nonlinear hyperbolic equations}, Israel Journal of Mathematics, 22 (1975), pp.~273--303. \bibitem {JJ} J. Shu, J. Zhang; \emph{Nonlinear Schr\"odinger equation with harmonic potential}, Journal of Mathematical Physics, 47, 063503 (2006). \bibitem{W} W. A. Strauss; \emph{Nonlinear wave equations}, Conference Board of the Mathematical Sciences, No. 73, American Mathematical Society, Providence, Rhode Island, 1989. \bibitem{MT} M. Tsutsumi; \emph{Nonexistence of global solutions to the Cauchy problem for nonlinear Schr\"odinger equations}, unpublished manuscript. \bibitem{TT3} T. Tsurumi, M. Wadati; \emph{Collapses of wave functions in multidimensional nonlinear Schr\"odinger equations under harmonic potential}, Journal of the Physical Society of Japan, 66 (1997), pp.~3031--3034. \bibitem {YJ} Y. Tsutsumi, J. Zhang; \emph{Instability of optical solitons for two-wave interaction model in cubic nonlinear media}, Advances in Mathematical Sciences and Applications , 8(1998), pp.~691--713. \bibitem{X1} R. Xu, Y. Liu; \emph{ Remarks on nonlinear Schr\"odinger equation with harmonic potential}, Journal of Mathematical Physics, 49, 043512 (2008). \bibitem{Y} K. Yajima; \emph{On fundamental solution of time dependent Schr\"odinger equations}, Contemporary Mathematics, 217 (1998), pp.~ 49--68. \bibitem{J1} J. Zhang; \emph{Sharp conditions of global existence for nonlinear Schr\"odinger and Klein--Gordon equations}, Nonlinear Analysis, 48 (2002), pp.~191--207. \bibitem{J2} J. Zhang; \emph{Stability of standing waves for nonlinear Schr\"odinger equations with unbounded potentials}, Zeitschrift f\"{u}r angewandte Mathematik und Physik, 51 (2000), pp.~498--503. \bibitem{J3} J. Zhang; \emph{Stability of attractive Bose--Einstein condensates}, Journal of Statistical Physics, 101 (2000), pp.~731--746. \bibitem{J4} J. Zhang; \emph{Cross-constrained variational problem and nonlinear Schr\"odinger equations}, Foundation of Computational Mathematics-Proceedings of the Smalefest, 2001, pp.~457--469. \end{thebibliography} \end{document}