\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2019 (2019), No. 83, pp. 1-22.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2019 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2019/83\hfil Fourth-order semilinear Schr\"odinger equations] {Difficulties in obtaining finite time blowup for fourth-order semilinear Schr\"odinger equations in the variational method frame} \author[R. Xu, Q. Lin, S. Chen, G. Wen, W. Lian \hfil EJDE-2019/83\hfilneg] {Runzhang Xu, Qiang Lin, Shaohua Chen, Guojun Wen, Wei Lian} \address{Runzhang Xu \newline College of Automation and College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China} \email{xurunzh@163.com} \address{Qiang Lin \newline College of Automation, Harbin Engineering University, Harbin 150001, China} \email{Linqiang\_edu@126.com} \address{Shaohua Chen \newline Department of Mathematics, Cape Breton University, Sydney, NS, B1P 6L2, Canada} \email{george\_chen@cbu.ca} \address{Guojun Wen \newline College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China} \email{964219363@qq.com} \address{Wei Lian \newline College of Automation, Harbin Engineering University, Harbin 150001, China} \email{lianwei\_1993@163.com} \thanks{Submitted January 11, 2019. Published June 24, 2019.} \subjclass[2010]{35B44, 35G25, 35A01, 35Q55} \keywords{Fourth-order Schr\"odinger equation; global solution; blowup; \hfill\break\indent variational problem; invariant manifolds} \begin{abstract} This article concerns the Cauchy problem for fourth-order semilinear Schr\"odinger equations. By constructing a variational problem and some invariant manifolds, we prove the existence of a global solution. Then we analyze the difficulties in proving the finite time blowup of the solution for the corresponding problem in the frame of the variational method. Understanding the finite time blowup of solutions, without radial initial data, still remains an open problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction}\label{sec1} In this article, we consider the Cauchy problem of the semilinear fourth-order Schr\"odinger equation \begin{equation}\label{x3.1} \begin{gathered} iu_t+\Delta u-\Delta^2u=-|u|^pu,\;(x,t)\in\mathbb{R}^N\times[0,T),\\ u(0,x)=u_0(x), \end{gathered} \end{equation} where $i=\sqrt{-1}$, $\Delta^2=\Delta\Delta$ is the biharmonic operator, $\Delta=\sum_{i=1}^{N}\frac{\partial^2}{\partial{x_i^2}}$ is the Laplace operator in $\mathbb{R}^N$; $u(x,t) :\mathbb{R}^N\times[0,T)\to \mathbb{C}$ denotes the complex valued function, $T$ is the maximum existence time; $N$ is the space dimension and $p$ satisfies the embedding condition \begin{equation} \label{eA} 0
4.
\end{cases}
\end{equation}
There has been a lot of interest in
fourth-order semilinear Schr\"odinger equation, because of their
strong physical background.
Karpmam \cite{zc0} investigated the fourth-order Schr\"odinger equation
\begin{equation}\label{az1}
iu_t+\frac{1}{2}\Delta u+\frac{1}{2}\gamma\Delta^2u+|u|^{2p}u=0,\,
\end{equation}
where $\gamma\in\mathbb{R}$, $p\geq1$, and the space dimension is no more than three.
Equation \eqref{az1} describes a stable soliton which is a wave pulse or wave beam,
specially, there are solitons in magnetic materials for $p=1$ in $3$-$D$ space.
Karpmam and Shagalov \cite{w6} presented a numerical study on the
axially symmetric fourth-order Schr\"odinger equation
\begin{equation}\label{az2}
i\frac{\partial u}{\partial\xi}+\frac{1}{2}S\Delta_{\bot}u
+\lambda\Delta^2_{\bot}u+\mu|u|^2u=0,
\end{equation}
where $S>0$, $\mu>0$,
$\Delta_{\bot}=\partial^2/\partial\rho^2+(1/\rho)\partial/\partial\rho$, and
$\xi, \rho$ are properly normalized cylindrical variables.
For $\lambda<0$, Equation \eqref{az2} plays a crucial role in the self-focusing,
here the fourth derivative term in \eqref{az2} may give rise to an
oscillatory approach to the asymptotically homogeneous wave beam.
Fibich et al.\ \cite{w10} analyzed the self-focusing and singularity formation
in the mixed-dispersion nonlinear Schr\"odinger equation
\begin{equation}\label{az3}
iu_t+\Delta u+\epsilon\Delta^2u+|u|^{2p}u=0,\quad (x,t)\in\mathbb{R}^N\times[0,T),
\end{equation}
where $\epsilon=\pm1$, $p\geq1$, which occurs
in propagation models for fiber arrays.
The authors showed that the generic propagation dynamics for $\epsilon<0$
is focusing-defocusing oscillations.
Davydova et al. \cite{w12} considered the Schr\"odinger equation in dimensionless
variables
\begin{equation}\label{az4}
iu_t+D\Delta u+P\Delta^2u+B|u|^2u+K|u|^4u=0,
\end{equation}
where $D, P, B, K\in\mathbb{R}$ and $BK<0$.
This equation was used for describing the dynamics of slowly varying wave
packet envelope amplitude.
Given its mathematical interests, a lot of attention is paid to the existence
and nonexistence of global solutions to the fourth-order
semilinear Schr\"odinger equation.
Pausader \cite{ww4} studied the equation
\begin{equation}
iu_t+\Delta^2u+\beta\Delta u+\lambda|u|^{p-1}u=0, \quad
(x,t)\in\mathbb{R}^N\times[0,T),
\end{equation}
where $\lambda, \beta\in\mathbb{R}$, $p\in (1,2^\#-1]$, and
$2^\#=\frac{2N}{N-4}$ is the energy critical exponent for the embedding from
$H^2$ into Lebesgue's spaces. Using the Strichartz-type estimates and
Gagliardo-Nirenberg's inequalities, he proved the local well-posedness,
and investigated global well-posedness and scattering with radially symmetrical
initial data.
Fibich et al.\ \cite{w10} proved the existence of a global solution
to the Cauchy problem
\begin{equation}\label{x0.02}
\begin{gathered}
iu_t+\epsilon\Delta^2 u+|u|^{2p}u=0,\quad (x,t)\in\mathbb{R}^N\times[0,T),\\
u(0,x)=u_0(x),
\end{gathered}
\end{equation}
under each of the following three sets of conditions:
(i) $\epsilon>0$,
(ii) $\epsilon<0$ and $pN<4$,
(iii) $\epsilon<0$, $pN=4$ and $||u_0||^2_2<\|R_B\|^2_2$,
where $R_B$ is the ground-state solution of
$-\Delta^2R_B-R_B+R_B^{\frac{8}{N}+1}=0$.
Furthermore, the authors gave the global well-posedness of problem \eqref{x0.02}
on a bounded domain $\Omega\subset\mathbb{R}^N$ with Dirichlet boundary
condition when (i)--(iii) are satisfied.
Based on the numerical simulations instead of rigorous mathematical proof,
the blowup solution was showed. And they figured out sufficient
conditions for existence of global solution for \eqref{x0.02} and for
\eqref{az3}.
Guo and Cui \cite{yf1} studied the Cauchy problem of the equation
\begin{equation}\label{jhbb0}
\begin{gathered}
iu_t+\mu\Delta^2u+\lambda\Delta u+f(|u|^2)u=0,\quad
(x,t)\in\mathbb{R}^{N}\times[0,T),\\
u(0,x)=u_0(x),
\end{gathered}
\end{equation}
where $\lambda\in\mathbb{R}$, $\mu\neq0$, and $f$ is a given real-valued
nonlinear function. Let $N=1, 2, 3$, by the standard contraction mapping argument,
a local solution for $u_0\in H^k$ and $k>\frac{N}{2}$ was obtained.
Then the authors obtained a global solution of \eqref{jhbb0} with $\nu u^{2p}$
instead of $f(|u|^2)$, for each of the following three sets of conditions:
(i) $\mu\nu>0$;
(ii) $\mu\nu<0$ and $0 0$, in the mass-supercritical case $\frac{4}{N} 0$ and $N\geq3$, $-|x|^{-2}|u|^{\frac{4}{N}}$ works as an
attractive self-reinforcing potential, $-|x|^{-2}|u|^{\frac{4}{N}}u$ is a
Hartree type nonlinearity, $\psi(x)$ is a sufficiently smooth and decreasing
function. They investigated existence and finite time blowup of local solution to
\eqref{tzx6} for negative initial energy $(E(u_0)<0)$.
In summary for the Cauchy problem \eqref{tzx8}
of the fourth-order Schr\"odinger equation with term $-\Delta^2u$, when
$u_0$ is radially symmetric and $E(u_0)<0$:
for the mass-critical case, the solution of \eqref{tzx8} either blows up in
finite time, or blows up in infinite time \cite{tzx3};
for the mass-supercritical and energy-critical cases, the solution of \eqref{tzx8}
blows up in finite time \cite{tzx3}; because of
the lack of conservation of mass, the solution of \eqref{tzx8} blows up in
finite time for $4/N 0$, the finite time blowup solution
of \eqref{tzx8} was proved in the energy-critical case and the mass-supercritical
case ($u_0$ is not necessarily radial) in \cite{tzx3}.
For the Cauchy problem \eqref{x3.1} of the fourth-order Schr\"odinger equation
with radial initial data, which contains both $-\Delta^2u$ and $\Delta u$,
$E(u_0)<0$ is currently a sufficient condition for the finite time blowup of
solution in mass-critical, mass-supercritical and energy-critical cases \cite{tzx3}.
The above discussions indicate that there is no blowup result for problem
\eqref{x3.1} when the initial energy is non-negative, i.e., $E(u_0)\geq 0$.
Hence we have no sharp condition for problem \eqref{x3.1} in positive initial
energy case, which even can be derived for the second-order nonlinear
Schr\"odinger equation \cite{jh010}. As the sharp condition is not only the
sufficient condition of blowup, but also its necessary condition to some extent,
and links the initial conditions of blowup and global existence,
we desire to obtain it for problem \eqref{x3.1},
similar to the case of second-order nonlinear Schr\"odinger equation as
follows:
(i) If $u_0\in\mathbb{B}:=\{u\in{H^1(\mathbb{R}^N)}:I(u)<0$, $E(u) 2.
\end{cases}
\]
Consider the Cauchy problem \eqref{3.9}, we define the energy functional
$$
\mathbf E(u)=\int_{\mathbb{R}^N}
\Big(\frac{1}{2}|\nabla u|^2-\frac{1}{p+2}|u|^{p+2}\Big)dx
$$
and the auxiliary functionals
\begin{gather*}
\mathbf{P}(u)=\int_{\mathbb{R}^N}\Big(\frac{1}{2}|u|^2
+\frac{1}{2}|\nabla u|^2-\frac{1}{p+2}|u|^{p+2}\Big)dx, \\
\intertext{and}
\mathbf{I}(u)=\int_{\mathbb{R}^N}
\Big(|u|^2+|\nabla u|^2-\frac{Np}{2(p+2)}|u|^{p+2}\Big)dx.
\end{gather*}
For the above two functionals, $\mathbf{P}(u)$ is composed of both mass and
energy, and $\mathbf{I}(u)$ can be considered as Nehari functional.
Further we define the Hilbert space
\[
\mathbf{H}=\{u\in H^1(\mathbb{R}^N):\int_{\mathbb{R}^N}|x|^2|u|^2dx<\infty\}\,,
\]
the Nehari manifold
\[
\mathbf{M}=\{u\in H^1(\mathbb{R}^N)\setminus\{0\}: \mathbf{I}(u)=0\}\,,
\]
and the invariant manifolds
\begin{gather*}
\mathbf{G}=\{u\in\mathbf{ H}: \mathbf{P}(u)<\mathbf{d},\mathbf{I}(u)>0\}
\cup\{0\} \\
\intertext{and}
\mathbf{B}=\{u\in\mathbf{ H}: \mathbf{P}(u)<\mathbf{d},\mathbf{I}(u)<0\},
\end{gather*}
where
\[
\mathbf{d}=\inf_{u\in\mathbf{M}}\mathbf{P}(u).
\]
\begin{remark}\label{NA1} \rm
(i) For set $\mathbf{G}$, we can obtain $\mathbf{P}(u)>0$ by $\mathbf{I}(u)>0$.
So the set $\mathbf{G}$ is equivalent to
$$
\mathbf{G'}=\left\{u\in\mathbf{ H}|0<\mathbf{P}(u)<\mathbf{d},\mathbf{I}(u)>0\right\}\cup\left\{0\right\}.
$$
(ii). For set $\mathbf{B}$, if $\mathbf{P}(u)\leq0$, we can get $\mathbf{E}(u)<0$, which is a sufficient condition for finite time blowup, cf. \cite{jh010}. Therefore, it is only necessary here to consider the case of $\mathbf{E}(u)>0$, i.e., we only need
$$
\mathbf{B'}=\left\{u\in\mathbf{ H}|0<\mathbf{P}(u)<\mathbf{d},\mathbf{I}(u)<0\right\}.
$$
\end{remark}
The above remark is also applicable to sets $G$ and $B$ in Section 3.
For the Cauchy problem \eqref{3.9} of the second-order semilinear Schr\"odinger
equation, we summarize some results established in \cite{w1,w5,www0.1,w20,w4,jh010}
as follows.
\begin{theorem}\label{thm1}
Assume that $u_0\in\mathbf{H}$ and $p$ satisfies the embedding condition
\[
\frac{4}{N} 2.
\end{cases}
\]
\begin{itemize}
\item[(i)]
(Local existence \cite{w1,w20})
There exists $T>0$ and a unique local solution $u(x,t)$ of problem \eqref{3.9}
in $C([0,T_{\rm max}];\mathbf{H})$. Moreover if
\[
T_{\rm max}=\sup \{T>0: u=u(x,t) \text{ exists on } [0,T]\}<\infty
\]
then
$$
\lim_{t\to T_{\rm max}}\|u\|_{\mathbf{H}}=\infty \quad \text{(blowup)},
$$
otherwise $T_{\rm max}=\infty$ (global existence).
\item[(ii)] (Conservation laws \cite{ w1,www0.1})
For the solution in (i),
\begin{gather*}
\int_{\mathbb{R}^N}|u(t)|^2
=\int_{\mathbb{R}^N}|u_0|^2dx\quad\text{(mass conservation)},\\
\mathbf E(u(t))= \mathbf{E}(u_0)\quad\text{(energy conservation)}, \\
\mathbf{P}(u(t))\equiv \mathbf{P}(u_0).
\end{gather*}
\item[(iii)]
$\mathbf{d}>0$, cf. \cite{w5,w4}.
\item[(iv)] (Global existence \cite{jh010})
If $u_0\in\mathbf{G}$, then the solution of problem \eqref{3.9} is global.
\item[(v)] (Blowup \cite{jh010})
If $u_0\in\mathbf{B}$, then the solution of problem \eqref{3.9} blows
up in finite time.
\end{itemize}
\end{theorem}
In fact, although \cite{jh010} proved the blowup solution by a variance of the
argument in \cite{www0.1}, there is no explicit computation of $\mathbf{J}''(t)$.
Now we give the specific computation of $\mathbf{J}''(t)$ for the Cauchy
problem \eqref{3.9}.
\begin{theorem}\label{wy1}
Assume that $u_0\in\mathbf{B}$, $u(x,t)\in([0,T);\mathbf{H})$ is the solution
of \eqref{3.9}. Let $\mathbf{J}(t)=\int_{\mathbb{R}^N}|x|^2|u|^2dx$, then
\[
\mathbf{J}''(t)=8\int_{\mathbb{R}^N}
\Big(|\nabla u|^2-\frac{Np}{2(p+2)}|u|^{p+2}\Big)dx.
\]
\end{theorem}
\begin{proof}
First
\begin{equation}\label{3.10}
\mathbf{J}'(t)=\int_{\mathbb{R}^N}|x|^2\left(u\bar{u}_t+\bar{u}u_t\right)dx
=\int_{\mathbb{R}^N}|x|^2\left(\overline{\bar{u}u_t}+\bar{u}u_t\right)dx
=2\operatorname{Re}\int_{\mathbb{R}^N}|x|^2\bar{u}u_t\,dx.
\end{equation}
Multiplying both sides of \eqref{3.9} by $i$, we have
\[
u_t=i\left(\Delta u+|u|^pu\right).
\]
Substituting the above equation into \eqref{3.10} we have
\begin{align*}
\mathbf{J}'(t)
=&2\operatorname{Re}\int_{\mathbb{R}^N}i|x|^2\bar{u}\left(\Delta u+|u|^pu\right)dx\\
=&-2\operatorname{Im}\int_{\mathbb{R}^N}|x|^2\bar{u}\left(\Delta u+|u|^pu\right)dx\\
=&-2\operatorname{Im}\int_{\mathbb{R}^N}|x|^2\left(\bar{u}\Delta u+|u|^{p+2}\right)dx\\
=&-2\operatorname{Im}\int_{\mathbb{R}^N}|x|^2\bar{u}\Delta udx\\
=&2\operatorname{Im}\int_{\mathbb{R}^N}|x|^2u\Delta \bar{u}dx,
\end{align*}
further
\begin{equation}\label{3.11}
\begin{aligned}
\mathbf{J}''(t)
=&2\operatorname{Im}\int_{\mathbb{R}^N}|x|^2\left(u_t\Delta\bar{u}+u\Delta\bar{u}_t\right)dx\\
=&2\operatorname{Im}\int_{\mathbb{R}^N}\left(|x|^2u_t\Delta\bar{u}+\Delta(|x|^2u)\bar{u}_t\right)dx\\
=&2\operatorname{Im}\int_{\mathbb{R}^N}
\Big(|x|^2u_t\Delta\bar{u}+\bar{u}_t\sum_{i=1}^N
\frac{\partial^2}{\partial x_i^2}(|x|^2u)\Big)dx\\
=&2\operatorname{Im}\int_{\mathbb{R}^N}
\Big(|x|^2u_t\Delta\bar{u}+\bar{u}_t\sum_{i=1}^N\frac{\partial}{\partial x_i}
\Big(|x|^2\frac{\partial u}{\partial x_i}+2x_iu\Big)\Big)dx\\
=&2\operatorname{Im}\int_{\mathbb{R}^N}
\Big(|x|^2u_t\Delta\bar{u}
+\bar{u}_t\Big(2Nu+4\sum_{i=1}^Nx_i\cdot\frac{\partial u}{\partial x_i}
+|x|^2\sum_{i=1}^N\frac{\partial^2 u}{\partial x_i^2}\Big)\Big)dx\\
=&2\operatorname{Im}\int_{\mathbb{R}^N}\left(|x|^2u_t\Delta\bar{u}+\bar{u}_t\left(2Nu+4x\cdot\nabla u+|x|^2\Delta u\right)\right)dx\\
=&2\operatorname{Im}\int_{\mathbb{R}^N}\left(|x|^2u_t\Delta\bar{u}+\overline{|x|^2u_t\Delta\bar{u}}+\bar{u}_t\left(2Nu+4x\cdot\nabla u\right)\right)dx\\
=&4\operatorname{Im}\int_{\mathbb{R}^N}\left(Nu+2x\cdot\nabla u\right)\bar{u}_t\,dx.
\end{aligned}
\end{equation}
According to \eqref{3.9}, we obtain
\begin{equation}
\bar{u}_t=-i(\Delta\bar{u}+|u|^p\bar{u}).
\end{equation}
Substituting the above equation into \eqref{3.11}, we can get
\begin{equation}\label{3.12}
\begin{aligned}
\mathbf{J}''(t)
=&-4\operatorname{Im}\int_{\mathbb{R}^N}i\left(Nu+2x\cdot\nabla u\right)\left(\Delta\bar{u}+|u|^p\bar{u}\right)dx\\
=&-4\operatorname{Re}\int_{\mathbb{R}^N}\left(Nu+2x\cdot\nabla u)(\Delta\bar{u}+|u|^p\bar{u}\right)dx\\
=&-4\Big(\operatorname{Re}\int_{\mathbb{R}^N}\left(Nu+2x\cdot\nabla u\right)
\Delta\bar{u}dx+\operatorname{Re}\int_{\mathbb{R}^N}
\left(Nu+2x\cdot\nabla u\right)|u|^p\bar{u}dx\Big)\\
=&-4(\mathbf{I_1}+\mathbf{I_2}),
\end{aligned}
\end{equation}
where
\[
\mathbf{I}_1:=\operatorname{Re}
\int_{\mathbb{R}^N}\left(Nu+2x\cdot\nabla u\right)\Delta\bar{u}dx\quad
\text{and}\quad
\mathbf{I}_2:=\operatorname{Re}\int_{\mathbb{R}^N}
\left(Nu+2x\cdot\nabla u\right)|u|^p\bar{u}dx.
\]
For $\mathbf{I}_1$ and $\mathbf{I}_2$, we have
\begin{equation}
\begin{aligned}
\mathbf{I}_1
=&N\operatorname{Re}\int_{\mathbb{R}^N}u\Delta\bar{u}dx+2\operatorname{Re}
\int_{\mathbb{R}^N}(x\cdot\nabla u)\Delta\bar{u}dx \\
=&-N\int_{\mathbb{R}^N}|\nabla u|^2dx-2\operatorname{Re}
\int_{\mathbb{R}^N}\nabla(x\cdot\nabla u)\nabla\bar{u}dx \\
=&-N\int_{\mathbb{R}^N}|\nabla u|^2dx
-2\operatorname{Re}\int_{\mathbb{R}^N}
\sum_{i=1}^N\frac{\partial}{\partial x_i}
\Big(\sum_{j=1}^Nx_j\frac{\partial u}{\partial x_j}\Big)
\frac{\partial\bar{u}}{\partial x_i}dx \\
=&-N\int_{\mathbb{R}^N}|\nabla u|^2dx-2\operatorname{Re}
\int_{\mathbb{R}^N}\sum_{i=1}^N\sum_{j=1}^N\frac{\partial}{\partial x_i}
\left(x_j\frac{\partial u}{\partial x_j}\right)
\frac{\partial\bar{u}}{\partial x_i}dx \\
=&-N\int_{\mathbb{R}^N}|\nabla u|^2dx-2\operatorname{Re}
\int_{\mathbb{R}^N}\sum_{i=1}^N\frac{\partial u}{\partial x_i}
\frac{\partial\bar{u}}{\partial x_i}dx \\
&-2\operatorname{Re}\int_{\mathbb{R}^N}\sum_{i=1}^N\sum_{j=1}^N
x_j\frac{\partial^2 u}{\partial x_i\partial x_j}
\frac{\partial\bar{u}}{\partial x_i}dx \\
=&-N\int_{\mathbb{R}^N}|\nabla u|^2dx-2\int_{\mathbb{R}^N}|\nabla u|^2dx \\
&-\operatorname{Re}\int_{\mathbb{R}^N}\sum_{i=1}^N\sum_{j=1}^Nx_j
\Big(\frac{\partial^2 u}{\partial x_i\partial x_j}
\frac{\partial\bar{u}}{\partial x_i}
+\frac{\partial^2\bar{u}}{\partial x_i\partial x_j}
\frac{\partial u}{\partial x_i}\Big)dx \\
=&-N\int_{\mathbb{R}^N}|\nabla u|^2dx-2\int_{\mathbb{R}^N}|\nabla u|^2dx
-\operatorname{Re}\int_{\mathbb{R}^N}\sum_{i=1}^N\sum_{j=1}^Nx_j
\frac{\partial}{\partial x_j}
\Big(\frac{\partial u}{\partial x_i}\frac{\partial\bar{u}}{\partial x_i}\Big)dx
\\
=&-N\int_{\mathbb{R}^N}|\nabla u|^2dx-2\int_{\mathbb{R}^N}|\nabla u|^2dx
-\operatorname{Re}\int_{\mathbb{R}^N}x\cdot\nabla|\nabla u|^2dx \\
=&-N\int_{\mathbb{R}^N}|\nabla u|^2dx
-2\int_{\mathbb{R}^N}|\nabla u|^2dx+N\int_{\mathbb{R}^N}|\nabla u|^2dx \\
=&-2\int_{\mathbb{R}^N}|\nabla u|^2dx
\end{aligned} \label{3.13}
\end{equation}
and
\begin{align*}
\mathbf{I}_2
=&N\int_{\mathbb{R}^N}|u|^{p+2}dx
+\operatorname{Re}\int_{\mathbb{R}^N}x\cdot
\left(|u|^p\left(\bar{u}\nabla u+u\nabla\bar{u}\right)\right)dx\\
=&N\int_{\mathbb{R}^N}|u|^{p+2}dx
+\operatorname{Re}\int_{\mathbb{R}^N}x\cdot
\left(|u|^p\nabla\left(u\bar{u}\right)\right)dx\\
=&N\int_{\mathbb{R}^N}|u|^{p+2}dx+\operatorname{Re}
\int_{\mathbb{R}^N}x\cdot\Big((|u|^2)^{p/2}\nabla|u|^2\Big)dx\\
=&N\int_{\mathbb{R}^N}|u|^{p+2}dx+\frac{2}{p+2}\operatorname{Re}
\int_{\mathbb{R}^N}x\cdot\nabla(|u|^2)^{\frac{p+2}{2}}dx\\
=&N\int_{\mathbb{R}^N}|u|^{p+2}dx-\frac{2N}{p+2}\operatorname{Re}
\int_{\mathbb{R}^N}|u|^{p+2}dx\\
=&\frac{Np}{p+2}\operatorname{Re}\int_{\mathbb{R}^N}|u|^{p+2}dx.
\end{align*}
Substituting the above equation and \eqref{3.13} into \eqref{3.12},
finally we derive
\begin{align*}
\mathbf{J}''(t)
=&-4\Big(-2\int_{\mathbb{R}^N}|\nabla u|^2dx+\frac{Np}{p+2}\operatorname{Re}
\int_{\mathbb{R}^N}|u|^{p+2}dx\Big)\\
=&8\int_{\mathbb{R}^N}\Big(|\nabla u|^2-\frac{Np}{2(p+2)}|u|^{p+2}\Big)dx.
\end{align*}
Then the proof is complete.
\end{proof}
\begin{remark}\rm
From the above structure of $\mathbf{J}''(t)$ and $\mathbf{I}(u)$, we easily
judge that $\mathbf{J}''(t)<0$ in the case of $\mathbf{I}(u)<0$, further the
blowup of solution for the second-order semilinear Schr\"odinger equation is derived.
\end{remark}
\section{Nonlinear fourth-order Schr\"odinger equation}
In this section, we consider the nonlinear fourth-order Schr\"odinger equation
for the Cauchy problem \eqref{x3.1}. First we define the space
\begin{equation}\label{x0.1}
H^2=\big\{u\in H^2(\mathbb{R}^N):\int_{\mathbb{R}^N}|x|^2|u|^2dx<\infty\big\},
\end{equation}
the energy functional
\begin{equation}\label{x0.4}
E(u(t))=\int_{\mathbb{R}^N}
\Big(\frac{1}{2}|\nabla u|^2+\frac{1}{2}|\Delta u|^2-\frac{1}{p+2}|u|^{p+2}\Big)dx
\end{equation}
and the auxiliary functionals
\begin{gather*}
P(u)=\int_{\mathbb{R}^N}\Big(\frac{1}{2}|u|^2+\frac{1}{2}|\nabla u|^2
+\frac{1}{2}|\Delta u|^2-\frac{1}{p+2}|u|^{p+2}\Big)dx, \\
\intertext{and}
I(u)=\int_{\mathbb{R}^N}\Big(|u|^2+|\nabla u|^2
+|\Delta u|^2-\frac{Np}{2(p+2)}|u|^{p+2}\Big)dx,
\end{gather*}
where $P(u)$ is composed of both mass and energy, and
$I(u)$ is considered as a Nehari functional.
The Nehari manifold is
\[
M=\{u\in H^2\setminus\{0\}:I(u)=0\}.
\]
Then we introduce the stable set $G$ and unstable set $B$:
\begin{gather*}
G=\{u\in H^2|P(u)