Cauchy problem of nonlinear Klein–Gordon equations with general nonlinearities

Abstract

This paper is concerned with the Cauchy problem of nonlinear Klein–Gordon equations with general nonlinearities. We use the potential well and convexity methods to prove the global existence and finite time blow up of solution with low and critical initial energy levels. And a finite time blow up of the solution with arbitrarily positive initial energy level is proved.

1 Introduction

In this paper, we consider the following nonlinear Klein–Gordon equation with general nonlinearities

$$\begin{aligned}&u_{tt}-\Delta u+ u=f(u), \quad x\in {\mathbb {R}}^n,\ t>0, \end{aligned}$$

(1.1)

$$\begin{aligned}&u(x,0)=u_0(x), \quad x\in {\mathbb {R}}^n, \end{aligned}$$

(1.2)

$$\begin{aligned}&u_{t}(x,0)=u_{1}(x),\quad x\in {\mathbb {R}}^n, \end{aligned}$$

(1.3)

where ${\mathbb {R}}^n (n\ge 1)$ is a unbounded domain, $\Delta$ is the Laplacian operator on ${\mathbb {R}}^n$,

$$\begin{aligned}&u_0\in H^1({\mathbb {R}}^n),\quad u_1\in L^2({\mathbb {R}}^n),\\&f(u)=a_{1}|u|^{p_{1}}+\sum _{k=2}^{l}a_{k}|u|^{p_{k}-1}u-\sum _{j=1}^{s}b_{j}|u|^{q_{j}-1}u,\nonumber \end{aligned}$$

(1.4)

and $a_{k}>0$, $1\le k\le l$, $b_{j}>0$, $1\le j\le s$, $a_{l+1}>0$, $p_{k}$ and $q_{j}$ satisfy

$$\begin{aligned} \mathrm{(H)} {\left\{ \begin{array}{ll} \mathrm{(1)}\ \text{ if }\ n=1,2, 1<q_{s}<q_{s-1}<\cdots<q_{1}=q<p=p_{1}<p_{2}<\cdots<p_{l}<\infty ;\\ \mathrm{(2)}\ \text{ if }\ n\ge 3, 1<q_{s}<q_{s-1}<\cdots<q_{1}=q<p=p_{1}<p_{2}<\cdots <p_{l}\le \frac{n+2}{n-2}. \end{array}\right. } \end{aligned}$$

The three-dimensional Klein–Gordon equation (1.1) was simplified as

$$\begin{aligned} \frac{1}{c^{2}}\frac{\partial ^{2}u}{\partial t^{2}}-\Delta u+V(u)u=0 \end{aligned}$$

to describe the quantum behavior of free particles [1]. For the one-dimensional Klein–Gordon equation

$$\begin{aligned} u_{tt}-\alpha u_{xx}+f(u)=0, \end{aligned}$$

(1.5)

when the nonlinear source $f(u)=\sin u$, $\sinh u$, $e^{u}$, $e^{u }+e^{-2u}$ or $e^{-u}+e^{-2u}$, the equations (1.5) are called Sine-Gordon equations, Sinh-Gordon equations, Liouville equations, Dodd–Bullough–Mikhailov equations or Tzitzeica–Dodd–Bullough equations respectively. Sine-Gordon equations and Sinh-Gordon equations are used to describe the propagation of fluxons in Josephson junction between two superconductors and the motion of a rigid pendulum attached to a stretched line, which is often seen in solid state physics and nonlinear optics [1,2,3]. Liouville equations are used to describe the vibration of uniformly charged plasma [4]. Dodd–Bullough–Mikhailov equations and Tzitzeica–Dodd–Bullough equations appear in various problems in fluids and quantum field theory [2, 5]. In addition, the Eq. (1.5) with nonlinear source $f(u)=-|u|^{2}u+|u|^{4}u$ are used to describe the quantum behavior of particles with spin 0 [6, 7].

In order to give a theoretical explanation from a mathematical point, we start from the well-posedness of the solution for the problem (1.1)–(1.3) to reveal the dynamic behavior of the solution. For the Cauchy problem of the Klein–Gordon equations (1.1)–(1.3) with $f(u)=|u|^{p-1}u$, Xu [9] proved the global existence and finite time blow up of the solution at low initial energy level $E(0)<d$. Li and Zhang [10] extended the result of finite time blow up and global existence to the critical initial energy level $E(0)=d$ for the Cauchy problem of the Klein–Gordon equation (1.1)–(1.3) with $f(u)=u^{2}+u^{3}$. Wang [8] proved the finite time blow up at the arbitrarily positive initial energy level $E(0)>0$ to the Cauchy problem of the Klein–Gordon equation (1.1)–(1.3) with $f(u)=u^{p-1}u$. Kutev et al. [11] considered Klein–Gordon equation (1.1)–(1.3) with $f(u)=\sum _{k=1}^{l}a_{k}|u|^{p_{k}-1}u-\sum _{j=1}^{s} b_{j}|u|^{q_{j}-1}u$ and $f(u)=a_{1}|u|^{p_{1}}+\sum _{k=2}^{l}a_{k}|u|^{p_{k}-1} u-\sum _{j=1}^{s}b_{j}|u|^{q_{j}-1}u$, where $a_{1}>0$, $a_{k}>0$, $2\le k\le l$, $b_{j}\ge 0$, $1\le j\le s$, $p_{k}$ and $q_{j}$ satisfy (H). The global existence and finite time blow up of the solution at the critical initial energy level $E(0)=d$ is proved in [12]. Kutev et al. [11] considered the same Cauchy problem and proved the finite time blow up of the solution at arbitrarily positive initial energy level $E(0)>0$.

As the nonlinear source terms reflect the influence of the nonlinear factors on the Cauchy problem of the Klein–Gordon equation (1.1)–(1.3), while the complex combined source terms describe the superimposition of these related factors. Obviously, the signs of the source terms greatly affect the size and direction of the combined source terms. The coefficients $a_{1}> 0$, $a_{k}>0$ and $b_{j}\ge 0$ in [11, 12], which mean that the directions of external force are positive, positive and negative respectively, and this nonlinear term is more general than the nonlinear term which appear in the damped semilinear wave equations [14], the fractional Laplacian parabolic [15], variable exponent parabolic equation [16], the parabolic systems [17], the pseudo-parabolic equation with singular potential [18], nonlinear Schröinger equation with a harmonic potential [19] and fractional p-Laplacian evolution equations [20]. In this present paper, we consider the nonlinear source $f(u)=a_{1}|u|^{p_{1}}+\sum _{k=2}^{l}a_{k} |u|^{p_{k}-1}u-\sum _{j=1}^{s}b_{j}|u|^{q_{j}-1}u$, where $a_{1 }<0$, $a_{k}>0$, $2\le k\le l$, $b_{j}\ge 0$, $1\le j\le s$, $p_{k}$ and $q_ {j}$ satisfy (H) to reveal the negative effects of the term $a_{1}|u|^{p_{1}}$, $a_{1 }<0$ and prove the well-posedness of the solution to the Cauchy problem of the Klein–Gordon equation (1.1)–(1.3).

2 Preliminaries

We commence this section by introducing the norms $\Vert u\Vert _{p}:=\Vert u\Vert _{L^{p}({\mathbb {R}}^{n})}$, $\Vert u\Vert :=\Vert u\Vert _{L^{2}({\mathbb {R}}^{n})}$ and the inner product $(u,v):=\int _{{\mathbb {R}}^{n}}uv\mathrm{d}x$. Also we introduce the norm for $H^{1}({\mathbb {R}}^{n})$

$$\begin{aligned} \Vert u\Vert _{H^{1}}^{2}:=\Vert u\Vert ^{2}+\Vert \nabla u\Vert ^{2}. \end{aligned}$$

We define two $C^{1}$ functionals on $H^{1}({\mathbb {R}}^{n})\rightarrow {\mathbb {R}}$, known as potential functional and Nehari function respectively as follows

$$\begin{aligned}&J(u):=\frac{1}{2}\Vert u\Vert _{H^{1}}^{2}-\sum _{k=2}^{l}\frac{a_{k}}{p_{k}+1}\Vert u\Vert _{p_{k}+1}^{p_{k}+1}-\frac{a_{1}}{p_{1}+1}\int _{{\mathbb {R}}^{n}}|u|^{p_{1}}u\mathrm{d}x+ \sum _{j=1}^{s}\frac{b_{j}}{q_{j}+1}\Vert u\Vert _{q_{k}+1}^{q_{k}+1} \end{aligned}$$

(2.1)

and

$$\begin{aligned} I(u):=\Vert u\Vert _{H^{1}}^{2}-\sum _{k=2}^{l}a_{k}\Vert u\Vert _{p_{k}+1}^{p_{k}+1}-a_{1}\int _{{\mathbb {R}}^{n}}|u|^{p_{1}}u\mathrm{d}x+ \sum _{j=1}^{s}b_{j}\Vert u\Vert _{q_{j}+1}^{q_{j}+1}. \end{aligned}$$

(2.2)

We also define Nehari manifold

$$\begin{aligned} {\mathcal {N}}:=\left\{ u\in H^{1}({\mathbb {R}}^{n})|\ I(u)=0, \Vert \nabla u\Vert \ne 0\right\} \end{aligned}$$

and the depth of the potential well (the so-called mountain pass level in [21])

$$\begin{aligned} d:=\inf \limits _{u\in {\mathcal {N}}}J(u). \end{aligned}$$

Now, we define the potential well

$$\begin{aligned} W:=\left\{ u\in H^{1}({\mathbb {R}}^{n})| I(u)>0, J(u)<d\right\} \cup \{0\}, \end{aligned}$$

the outer of the potential well

$$\begin{aligned} V:=\left\{ u\in H^{1}({\mathbb {R}}^{n})| I(u)<0, J(u)<d\right\} \end{aligned}$$

and the family of potential wells

$$\begin{aligned} I_{\delta }(u):=\delta \Vert u\Vert _{H^{1}}^{2}-a_{1}\int _{{\mathbb {R}}^{n}}|u|^{p_{1}}u\mathrm{d}x-\sum _{k=2}^{l}a_{k}\Vert u\Vert _{p_{k}+1}^{p_{k}+1}+ \sum _{j=1}^{s}b_{j}\Vert u\Vert _{q_{j}+1}^{q_{j}+1},\ \delta >0. \end{aligned}$$

The corresponding Nehari manifolds and the depth of the family of potential wells are defined respectively by

$$\begin{aligned} {\mathcal {N}}_{\delta }:=\left\{ u\in H^1({\mathbb {R}}^n)\big | I_{\delta }(u)=0, \Vert \nabla u\Vert \ne 0 \right\} \end{aligned}$$

and

$$\begin{aligned} d(\delta ):=\inf _{u\in {\mathcal {N}}_{\delta }} J(u). \end{aligned}$$

(2.3)

Next we introduce the stable set $W_{\delta }$ and the unstable set $V_{\delta }$ defined by

$$\begin{aligned} W_\delta := \{u\in H^1({\mathbb {R}}^n)\big | I_\delta (u)>0, J(u)<d(\delta )\}\cup \{0\} \end{aligned}$$

and

$$\begin{aligned} V_\delta :=\{u\in H^1({\mathbb {R}}^n)\big | I_\delta (u)<0,\ J(u)<d(\delta )\}. \end{aligned}$$

Definition 2.1

Function u(x, t) is called a week solution to problem (1.1)–(1.3), if it satisfies

$$\begin{aligned} u\in L^{\infty }\left( [0,T);H^1({\mathbb {R}}^n)\right) , u_t\in L^{\infty }\left( [0,T);L^2({\mathbb {R}}^n)\right) \end{aligned}$$

and there holds

$$\begin{aligned} \langle u_{tt},v\rangle +(\nabla u,\nabla v)+(u,v)=(f(u),v) \end{aligned}$$

(2.4)

for any $v\in H^{1}({\mathbb {R}}^n)$, $t\in [0,T)$, where $\langle \cdot , \cdot \rangle$ denotes the duality pairing between $H^{-1}({\mathbb {R}}^{n})$ and $H_{0}^{1}({\mathbb {R}}^{n})$, and the following energy equality holds

$$\begin{aligned} E(t)=E(0),\ t\in (0,T), \end{aligned}$$

(2.5)

where

$$\begin{aligned} E(t):=\frac{1}{2}\Vert u_t\Vert ^2+J(u). \end{aligned}$$

(2.6)

Lemma 2.2

Assume f(u) satisfy (H), then we have

$$\begin{aligned} (p+1)F(u) \le u f(u) \text{ for } \text{ all }\ u\in {\mathbb {R}}, \end{aligned}$$

(2.7)

where $F(u):=\int _0^u f(s)\mathrm{d}s$.

Proof

When $u\ge 0$, then by (i) in (H), we have

$$\begin{aligned} uf'(u)\ge pf(u) \end{aligned}$$

and

$$\begin{aligned} \int _{0}^{u} sf'(s)\mathrm{d}s\ge p\int _0^u f(s)\mathrm{d}s=pF(u)\quad \text {for all}\quad u\ge 0, \end{aligned}$$

which implies

$$\begin{aligned} uf(u)-\int _{0}^{u}f(s)\mathrm{d}s\ge pF(u)\quad \text {for all}\quad u\ge 0, \end{aligned}$$

that is

$$\begin{aligned} (p+1)F(u)\le u f(u). \end{aligned}$$

The proof of the case $u<0$ is similar to the case $u\ge 0$. $\square$

Lemma 2.3

Let f(u) satisfy (H), then

• (i) $|F(u)|\le \sum _{k=1}^{l}\frac{a_k}{p_k+1}|u|^{p_k+1}+\sum _{j=1}^{s}\frac{b_{j}}{q_{j}+1}|u|^{q_j+1}$ for all $u\in {\mathbb {R}}$;

• (ii) $F(u)\ge B|u|^{p+1}$ for $B=F(1)$ and all $|u|\ge 1$.

Proof

(i) From (H), it implies that

$$\begin{aligned} |F(u)|\le&\left| \int _{0}^{u}\left( \sum _{k=1}^{l}a_{k}|u|^{p_{k}-1}u+a_{1}|u|^{p_{1}}-\sum _{j=1}^{s}b_{j}|u|^{q_{j}-1}u\right) \mathrm{d}u\right| \\ \le&\sum _{k=1}^{l}\frac{a_k}{p_k+1}|u|^{p_k+1}+\sum _{j=1}^{s}\frac{b_{j}}{q_{j}+1}|u|^{q_j+1}. \end{aligned}$$

(ii) When $u>0$, by $\mathrm (H)$, we have $F(u)>0$. Then Lemma 2.2 tells

$$\begin{aligned} \frac{p+1}{u}\le \frac{f(u)}{F(u)}=\frac{\frac{\mathrm{d}F(u)}{\mathrm{d}u}}{F(u)}, \end{aligned}$$

which gives

$$\begin{aligned} (p+1)\int _1^u{\frac{1}{s}\mathrm{d}s}\le \int _{1}^{u}{\frac{\mathrm{d}F(s)}{F(s)}}, \end{aligned}$$

that is

$$\begin{aligned} \ln u^{p+1}\le \ln F(u)-\ln F(1). \end{aligned}$$

Then $F(u)\ge B u^{p+1},$ $B=F(1)$. Similarly to the case $u\ge 1$, we obtain $F(u)\ge Bu^{p+1}$ for all $u\le -1$. $\square$

Lemma 2.4

(Relations between I(u) and $\Vert \nabla u\Vert$) Let $\delta >0$.

• (1) If $0<\Vert u\Vert _{H^{1}}<\gamma (\delta )$, then $I_{\delta }(u)>0$. In particular, if $0<\Vert u\Vert _{H^{1}}<\gamma (1)$, then $I(u)>0;$

• (2) If $I_{\delta }(u)<0,$ then $\Vert u\Vert _{H^{1}}>\gamma (\delta ).$ In particular, if $I(u)<0,$ then $\Vert u\Vert _{H^{1}}>\gamma (1);$

• (3) If $I_{\delta }(u)=0$ and $\Vert u\Vert _{H^{1}}\ne 0$, then $\Vert u\Vert _{H^{1}}\ge \gamma (\delta )$. In particular, if $I(u)<0$ and $\Vert u\Vert _{H^{1}}\ne 0$, then $\Vert u\Vert _{H^{1}}\ge \gamma (1)$,

where $\gamma (\delta )$ is the unique real root of equation $\varphi (\gamma )=\delta$,

$$\begin{aligned} \varphi (\gamma ):=\sum _{k=1}^{l+1}a_{k}C_{k}^{p_{k}+1}\gamma ^{p_{k}-1},\ C_{k}:=\sup \limits _{u\in H_{0}^{1}({\mathbb {R}}^{n})}\frac{\Vert u\Vert _{p_{k}+1}}{\Vert u\Vert _{H^{1}}}, 1\le k\le l. \end{aligned}$$

Proof

(1) From $0<\Vert u\Vert _{H^{1}}\le \gamma (\delta )$, we have $\Vert u\Vert _{q_{j}+1}>0$, $1\le j\le s$. Hence by

$$\begin{aligned}&\sum _{k=2}^{l}a_{k}\Vert u\Vert _{p_{k}+1}^{p_{k}+1}+a_{1}\int _{{\mathbb {R}}^{n}}|u|^{p_{1}}u\mathrm{d}x-\sum _{j=1}^{s}b_{j}\Vert u\Vert _{q_{j}+1}^{q_{j}+1}\\&\quad <\sum _{k=1}^{l}a_{k}\Vert u\Vert _{p_{k}+1}^{p_{k}+1}\le \sum _{k=1}^{l} a_{k} C_{k}^{p_{k}+1}\Vert u\Vert _{H^{1}}^{p_{k}+1}\\&\quad =\varphi (\Vert u\Vert _{H^{1}})\Vert u\Vert _{H^{1}}^{2}\le \delta \Vert u\Vert _{H^{1}}^{2}, \end{aligned}$$

we get $I_{\delta }(u)>0$.

(2) The inequality $I_{\delta }(u)<0$ gives

$$\begin{aligned} \delta \Vert u\Vert _{H^{1}}^{2}&<\sum _{k=1}^{l}a_{k}\Vert u\Vert _{p_{k}+1}^{p_{k}+1}-\sum _{j=1}^{s}b_{j}\Vert u\Vert _{q_{j}+1}^{q_{j}+1}\nonumber \\&<\sum _{k=1}^{l}a_{k}\Vert u\Vert _{p_{k}+1}^{p_{k}+1}\le \varphi (\Vert u\Vert _{H^{1}})\Vert u\Vert _{H^{1}}^{2}, \end{aligned}$$

which implies $\Vert u\Vert _{H^{1}}>\gamma (\delta )$.

(3) If $I_{\delta }(u)=0$ and $\Vert u\Vert _{H^{1}}\ne 0$, then by

$$\begin{aligned} \delta \Vert u\Vert _{H^{1}}^{2}&=\sum _{k=1}^{l}a_{k}\Vert u\Vert _{p_{k}+1}^{p_{k}+1}-\sum _{j=1}^{s}b_{j}\Vert u\Vert _{q_{j}+1}^{q_{j}+1}\\&<\sum _{k=1}^{l}a_{k}\Vert u\Vert _{p_{k}+1}^{p_{k}+1}\le \varphi (\Vert u\Vert _{H^{1}})\Vert u\Vert _{H^{1}}^{2}, \end{aligned}$$

we get $\Vert u\Vert _{H^{1}}>\gamma (\delta )$. $\square$

Lemma 2.5

Let f(u) satisfy (H), $u\in H^1({\mathbb {R}}^n)$, $\Vert u\Vert _{H^1}\ne 0$ and

$$\begin{aligned} \varphi (\lambda ):=\frac{1}{\lambda }\int _{{\mathbb {R}}^{n}} uf(\lambda u)\mathrm{d}x. \end{aligned}$$

Then

• (i) $\varphi (\lambda )$ is strictly increasing on $[0,+\infty )$;

• (ii) $\lim \limits _{\lambda \rightarrow 0}\varphi (\lambda )=0,\ \lim \limits _{\lambda \rightarrow +\infty }\varphi (\lambda )=+\infty .$

Proof

(i) For $\lambda >0$, the conclusion follow from

$$\begin{aligned} \frac{\mathrm{d}\varphi (\lambda )}{\mathrm{d}\lambda }=&\frac{1}{\lambda }\int _{{\mathbb {R}}^n}u^2f'(\lambda u)\mathrm{d}x-\frac{1}{\lambda ^2}\int _{{\mathbb {R}}^n}uf(\lambda u)\mathrm{d}x\\ =&\frac{1}{\lambda ^3}\int _{{\mathbb {R}}^n}\lambda u\left( \lambda uf'(\lambda u)-f(\lambda u)\right) \mathrm{d}x>0 \end{aligned}$$

and (i) in (H).

(ii) It follows directly from (ii) in (H) that

$$\begin{aligned} |\varphi (\lambda )|\le&\frac{1}{\lambda ^2}\int _{{\mathbb {R}}^n}|\lambda uf(\lambda u)|\mathrm{d}x \le \frac{1}{\lambda ^2}\int _{{\mathbb {R}}^n}\sum _{k=1}^{l}a_k|\lambda u|^{p_k+1}\mathrm{d}x= \sum _{k=1}^{l}a_k\lambda ^{p_k-1}\Vert u\Vert _{p_k+1}^{p_k+1}, \end{aligned}$$

which implies that $\lim \limits _{\lambda \rightarrow 0}\varphi (\lambda )=0.$ From Lemma 2.2 and (ii) in Lemma 2.3, we get

$$\begin{aligned} \varphi (\lambda )=&\frac{1}{\lambda ^2}\int _{{\mathbb {R}}^n}|\lambda uf(\lambda u)|\mathrm{d}x\ge \frac{p+1}{\lambda ^2}\int _{{\mathbb {R}}_\lambda ^n}F(\lambda u)\mathrm{d}x\nonumber \\ \ge&\frac{p+1}{\lambda ^2}\int _{{\mathbb {R}}_\lambda ^n}B|\lambda u|^{p+1}\mathrm{d}x =(p+1)B\lambda ^{p-1}\int _{{\mathbb {R}}_\lambda ^n}|u|^{p+1}\mathrm{d}x, \end{aligned}$$

where ${\mathbb {R}}_\lambda ^n=\left\{ x|x\in {\mathbb {R}}^n, |u|\ge \frac{1}{\lambda }\right\} .$ Hence from

$$\begin{aligned} \lim _{\lambda \rightarrow +\infty }\int _{{\mathbb {R}}_\lambda ^n}|u|^{p+1}\mathrm{d}x=\Vert u\Vert _{p+1}^{p+1}>0, \end{aligned}$$

we have

$$\begin{aligned} \lim _{\lambda \rightarrow +\infty }\varphi (\lambda )=+\infty . \end{aligned}$$

$\square$

Lemma 2.6

(Properties of $J(\lambda u)$, Lemma 2.2 in [23] and Lemma 6 in [24]) Let $u\in H_{0}^{1}(\Omega )$ and $\Vert u\Vert _{H^{1}}\ne 0$. Then

• (i) $\lim \nolimits _{\lambda \rightarrow 0} J(\lambda u)=0$, $\lim \nolimits _{\lambda \rightarrow +\infty } J(\lambda u)=-\infty$.

• (ii) there exists a unique $\lambda ^{*}\in (0,\infty )$ such that

  $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\lambda }J(\lambda u)\bigg {|}_{\lambda =\lambda ^{*}}=0. \end{aligned}$$

• (iii) $J(\lambda u)$ is increasing on $0\le \lambda \le \lambda ^{*}$, decreasing on $\lambda ^{*}\le \lambda <\infty$ and takes the maximum at $\lambda =\lambda ^{*}$.

• (iv) $I(\lambda u)>0$ for $0<\lambda <\lambda ^{*}$, $I(\lambda u)<0$ for $\lambda ^{*}<\lambda <\infty$ and $I(\lambda ^{*} u)=0$.

Lemma 2.7

(Properties of $d(\delta )$, Lemma 3 in [25]) $d(\delta )$ possesses the following properties

• (i) $d(\delta )>a(\delta )\gamma ^{2}(\delta )$ for $0<\delta <\frac{p+1}{2}$, where $a(\delta ):=\frac{1}{2}-\frac{\delta }{p+1}$.

• (ii) $\lim \limits _{\delta \rightarrow 0}d(\delta )=d(0)$ and there exists a unique $b>\frac{p+1}{2}$ such that $d(\delta _{0})=0$ and $d(\delta )>0$ for $0\le \delta <b$.

• (iii) $d(\delta )$ is increasing on $0\le \delta \le 1$, decreasing on $1\le \delta \le \delta _{0}$ and takes the maximum d=d(1) at $\delta =1$.

Lemma 2.8

(Invariance sets for $E(0)<d$) Let f(u) satisfy (H), $u_0(x)\in H^1({\mathbb {R}}^n)$, $u_1(x)\in L^2({\mathbb {R}}^n)$. Suppose that $0\le e<d$, $\delta _1$ and $\delta _2$ are the two roots of equation $d(\delta )=e$. Then,

• (i) the solution of problem (1.1)–(1.3) with $0<E(0)\le e$ belongs to $W_\delta$ for $\delta _1<\delta <\delta _2$, provided that $I(u_0)>0$ or $\Vert u_0\Vert _{H^1}=0$;

• (ii) the solution of problem (1.1)–(1.3) with $0<E(0)\le e$ belongs to $V_\delta$ for $\delta _1<\delta <\delta _2$, provided that $I(u_0)<0$.

Proof

Assume $u=u(t)$ is the solution to problem (1.1)–(1.3) with $0<E(0)\le e$, $I(u_0)>0$ or $\Vert u_0\Vert _{H^1}=0$ and T is the maximum existence time of u(t). If $\Vert u_{0}\Vert _{H^1}=0$, then $u_{0}\in W_{\delta }$ for $\delta \in (\delta _{1},\delta _{2})$. If $I(u_{0})>0$, then by (2.3) and

$$\begin{aligned} \frac{1}{2}\Vert u_{1}\Vert +J(u_{0})=E(0)=d(\delta _{1})=d(\delta _{2})<d(\delta ),\ \delta \in (\delta _{1}, \delta _{2}), \end{aligned}$$

it follows that $I_{\delta }(u_{0})>0$ and $J(u_{0})<d(\delta )$, i.e., $u_{0}(x)\in W_{\delta }$ for $\delta \in (\delta _{1},\delta _{2})$. Next we prove $u(t)\in$ $W_\delta$ for $\delta \in (\delta _1,\delta _2)$, $t\in (0,T).$ Arguing by contradiction, we suppose that there exists a $t_0\in (0,T)$ such that $u(t_0)\in \partial W_\delta$, i.e.,

$$\begin{aligned} I_\delta (u_0)=0,\ \Vert u(t_0)\Vert _{H^1}\ne 0\quad \text{ and }\quad J(u(t_0))=d(\delta ),\ \delta \in (\delta _1,\delta _2). \end{aligned}$$

From (2.5), we obtain

$$\begin{aligned} \frac{1}{2}\Vert u_t\Vert ^2+J(u)=E(0)<d(\delta ),\ \delta \in (\delta _1,\delta _2),\ t\in (0,T), \end{aligned}$$

(2.8)

which implies that $J(u(t_0))=d(\delta )$ is impossible. If $I_\delta (u(t_0))=0$ and $\Vert u(t_0)\Vert _{H^1}\ne 0$, by (2.3), it follows that $J(u(t_0))\ge d(\delta )$, which contradicts (2.8). Similarly, we can achieve the second statement. $\square$

3 Global solution and finite time blow up for $E(0)<d$

Theorem 3.1

(Global existence for $E(0)<d$) Let f(u) satisfy (H), $u_0(x)\in H^1({\mathbb {R}}^n)$ and $u_1(x)\in L^2({\mathbb {R}}^n)$. Suppose that $E(0)<d$ and $I(u_0)>0$ or $\Vert u_0\Vert _{H^1}=0$. Then problem (1.1)–(1.3) admits a global weak solution

$$\begin{aligned} u(t)\in L^{\infty }\left( (0,\infty ); H^1({\mathbb {R}}^n)\right) \end{aligned}$$

with

$$\begin{aligned} u_t(t)\in L^{\infty }\left( (0,\infty );L^2({\mathbb {R}}^n)\right) \end{aligned}$$

and $u(t)\in W$ for $t\in (0,+\infty )$.

Proof

Let $\{\omega _{j}(x)\}$ be a system of base functions in $H_{0}^{1}({\mathbb {R}}^{n})$. Construct approximate solutions of problem (1.1)–(1.3) as

$$\begin{aligned} u_{m}(x,t)=\sum _{j=1}^{m}g_{jm}(t)\omega _{j}(x), m=1,2,\cdots \end{aligned}$$

satisfying

$$\begin{aligned}&\langle u_{mtt}, \omega _{s}\rangle +(\nabla u_{m}, \nabla \omega _{s})+(u_{m},\omega _{s})=(f(u_{m}), \omega _{s}),\quad s=1,2,\ldots , m,\\&u_{m}(x,0)=\sum _{j=1}^{m}g_{jm}(0)\omega _{j}(x)\rightarrow u_{0}(x)\in H_{0}^{1}({\mathbb {R}}^{n}) \end{aligned}$$

and

$$\begin{aligned} u_{mt}(x,0)=\sum _{j=1}^{m}g'_{jm}(0)\omega _{j}(x)\rightarrow u_{1}(x)\in L^{2}({\mathbb {R}}^{n}). \end{aligned}$$

Then by the same arguments used in the proof of Theorem 3.2 in [13], for sufficiently large m and $t\in (0,+\infty )$ we obtain

$$\begin{aligned} \frac{1}{2}\Vert u_{mt}\Vert ^{2}+J(u_{m})=E_{m}(0)<d \end{aligned}$$

(3.1)

and $u_{m}(t)\in W$. From (3.1) and

$$\begin{aligned} J(u_{m})&\ge \frac{1}{2}\Vert u_{m}\Vert _{H^{1}}^{2}-\frac{1}{p+1}\left( \sum _{k=2}^{l}a_{k}\Vert u\Vert _{p_{k}+1}^{p_{k}+1}+\int _{{\mathbb {R}}^{n}}|u|^{p_{1}}u\mathrm{d}x-\sum _{j=1}^{s}b_{j}\Vert u\Vert _{q_{k}+1}^{q_{k}+1}\right) \nonumber \\&=\left( \frac{1}{2}-\frac{1}{p+1}\right) \Vert u_{m}\Vert _{H^{1}}^{2}+\frac{1}{p+1}I(u_{m})\ge \frac{p-1}{2(p+1)}\Vert u_{m}\Vert _{H^{1}}^{2}, \end{aligned}$$

(3.2)

for sufficiently large m it follows that

$$\begin{aligned} \frac{1}{2}\Vert u_{mt}\Vert ^{2}+\frac{p-1}{2(p+1)}\Vert u_{m}\Vert _{H^{1}}^{2}<d, \ t\in (0,+\infty ), \end{aligned}$$

(3.3)

for sufficiently large m which implies that

$$\begin{aligned} \Vert u_{m}\Vert _{H^{1}}^{2}<\frac{2(p+1)}{p-1}d,\ t\in (0,+\infty ) \end{aligned}$$

(3.4)

and

$$\begin{aligned} \Vert u_{mt}\Vert ^{2}<2d,\ t\in (0,+\infty ). \end{aligned}$$

(3.5)

From the definition of $C_{k}$ and (3.3), for sufficiently large m we have

$$\begin{aligned} \Vert u_{m}\Vert _{p_{k}+1}^{2}\le C_{k}^{2}\Vert u_{m}\Vert _{H^{1}}^{2}<C_{k}^{2}\frac{2(p+1)}{p-1}d,\ 1\le k\le l,\ t\in (0,+\infty ). \end{aligned}$$

(3.6)

From (3.4)–(3.6) and compactness method it follows that problem (1.1)–(1.3) admits a global weak solution $u(t)\in L^{\infty }\left( [0,\infty ); H_{0}^{1}({\mathbb {R}}^{n})\right)$ with $u_{t}(t)\in L^{\infty }\left( [0,\infty ); L^{2}({\mathbb {R}}^{n})\right)$. Finally by Lemma 2.8, for $t\in (0,+\infty )$ we have $u(t)\in W$. $\square$

Theorem 3.2

(Finite time blow up for $E(0)<d$) Let f(u) satisfy (H), $u_0(x)\in H^1({\mathbb {R}}^n)$, $u_1(x)\in L^2({\mathbb {R}}^n)$. Assume that $E(0)<d$ and $I(u_0)<0$, then the solution to problem (1.1)–(1.3) blows up in finite time.

Proof

Arguing by contradiction, we assume the maximum existence time $T=+\infty$. First, for any $T>0$ we define

$$\begin{aligned} M(t):=\Vert u(t)\Vert ^2, \end{aligned}$$

(3.7)

then

$$\begin{aligned} M'(t)=2(u_t,u) \end{aligned}$$

(3.8)

and

$$\begin{aligned} M''(t)=2\Vert u_t\Vert ^2-2I(u) \end{aligned}$$

(3.9)

due to (2.4). From (3.1) and (3.2), we have

$$\begin{aligned} 2I(u)\le 2(p+1)E(0)-(p+1)\Vert u_t\Vert ^2-(p-1)\Vert u\Vert _{H^1}^2. \end{aligned}$$

(3.10)

Substituting (3.10) into (3.9), we obtain

$$\begin{aligned} M''(t)\ge (p-1)M(t)+(p+3)\Vert u_t\Vert ^2-2(p+1)E(0). \end{aligned}$$

(3.11)

Now, we consider the following two cases respectively.

(i) If $0<E(0)<d$, then from Lemma 2.8, it follows that $u(t)\in V_\delta$ for $1<\delta <\delta _2$ and $t>0$, where $\delta _2$ is the same as that in Theorem 2.8. Thus $I_\delta (u)<0$ and $\Vert u\Vert _{H^1}>\gamma (\delta )$ for $1<\delta <\delta _2$ and $t>0$. Therefore, we obtain $I_{\delta _2}(u)\le 0$ and $\Vert u\Vert _{H^1}\ge \gamma (\delta _2)$ for $t>0$ and by (3.9), for $t\in [0,T)$ we have

$$\begin{aligned} M''(t)\ge -2I(u)=&2(\delta _2-1)\Vert u\Vert ^2_{H^1}-2I_{\delta _2}(u)\nonumber \\ \ge&2(\delta _2-1)\gamma ^2(\delta _2)>0, \end{aligned}$$

then

$$\begin{aligned} M'(t)\ge&2(\delta _2-1)\gamma ^2(\delta _2)t+M'(0), \end{aligned}$$

which shows that there exists a $t_0\ge 0$ such that

$$\begin{aligned} M'(t)>M'(t_0)>0 \end{aligned}$$

and

$$\begin{aligned} M(t)\ge M'(t_0)(t-t_0)+M(t_0),\ t\ge t_0. \end{aligned}$$

Hence for a sufficiently large t, we get

$$\begin{aligned} (p-1)M(t)>2(p+1)E(0) \end{aligned}$$

and

$$\begin{aligned} M''(t)>(p+3)\Vert u_t\Vert ^2. \end{aligned}$$

(3.12)

Finally, Schwarz inequality tells

$$\begin{aligned} M(t)M''(t)-\frac{p+3}{4}(M'(t))^2 \ge M(t)\left( M''(t)-(p+3)\Vert u_t\Vert ^2\right) >0, \end{aligned}$$

then

$$\begin{aligned} (M^{-\alpha }(t))''=\frac{-\alpha }{M(t)^{\alpha +2}}\left( M(t)M''(t)-(\alpha +1)(M'(t))^2\right) <0,\ \alpha =\frac{p-1}{4} \end{aligned}$$

and

$$\begin{aligned} \lim _{t\rightarrow T^{*}} M(t)=+\infty \end{aligned}$$

for some $T^{*}>0$, which contradicts $T=+\infty .$

(ii) When $E(0)\le 0$, by (3.11), we obtain (3.12). The remaining proof is similar to the case (i). $\square$

4 Global existence and finite time blow up for $E(0)=d$

Theorem 4.1

(Global existence for $E(0)=d$) Let f(u) satisfy (H), $u_0(x) \in H^1({\mathbb {R}}^n)$, $u_1(x) \in L^2({\mathbb {R}}^n)$. Suppose that $E(0)=d$ and $u_0\in W$, then problem (1.1)–(1.3) admits a global weak solution $u(t) \in L^{\infty }\left( [0,T); H^1({\mathbb {R}}^n)\right)$ with $u_t(t) \in L^{\infty }\left( [0,T); L^2({\mathbb {R}}^n)\right)$.

Proof

We prove this theorem considering two cases (i) and (ii).

(i) In the case $\Vert u_0\Vert _{H^1}\ne 0,$ let $\lambda _m=1-\frac{1}{m}$ and $u_{0m}=\lambda _mu_0$, $m=2,3,\cdots$. Consider the initial conditions

$$\begin{aligned} u(x,0)=u_{0m}(x),\quad u_t(x,0)=u_1(x) \end{aligned}$$

(4.1)

and the corresponding problem (1.1), (4.1). From $I(u_0)>0$ and (iii), (iv) in Lemma 2.6, we have

$$\begin{aligned} \lambda ^*=\lambda ^*(u_0)\ge 1\ \text{ and }\ J(u_{0m})=J(\lambda _mu_0)<J(u_0). \end{aligned}$$

(4.2)

From (2.1), we replace $\Vert u_{0m}\Vert _{L_{p+1}^{\frac{n}{p+1}}({\mathbb {B}})}^{p+1}$ by $I(u_{0m})$ to have

$$\begin{aligned} J(u_{0m})\ge&\frac{1}{2}\Vert \nabla u_{0m}\Vert ^2+\frac{1}{2}\Vert u_{0m}\Vert ^2-\frac{1}{p+1}\int _{{\mathbb {R}}^n}u_{0m}f(u_{0m})\mathrm{d}x\nonumber \\ =&\frac{p-1}{2(p+1)}\Vert u_{0m}\Vert _{H^1}^2+\frac{1}{p+1}I(u_{0m})>0. \end{aligned}$$

(4.3)

From (4.2) and (4.3), we have

$$\begin{aligned} 0<E_m(0)\equiv J(u_{0m})+ \frac{1}{2}\Vert u_1\Vert ^2<\frac{1}{2}\Vert u_1\Vert ^2+J(u_0)=E(0)=d. \end{aligned}$$

Similar to the proof of Theorem 3.1, we finish this proof.

(ii) We discuss the case $\Vert u_m\Vert ^2_{H^1}=0$, which implies $J(u_0)=0$ and $\frac{1}{2}\Vert u_1\Vert ^2=E(0)=d$. Let $\lambda _m=1-\frac{1}{m}$, $u_{1m}(x)=\lambda _mu_1(x)$, $m=2,3,\ldots .$ We take initial conditions

$$\begin{aligned} u(x,0)=u_0(x),\ u_t(x,0)=u_{1m}(x) \end{aligned}$$

(4.4)

and consider the corresponding problem (1.1) and (4.4). From $J(u_{0})=0$ and (2.6), we have

$$\begin{aligned} 0<E_m(0)=\frac{1}{2}\Vert u_{1m}\Vert ^2+J(u_0)=\frac{1}{2}\Vert \lambda _mu_1\Vert ^2<\frac{1}{2}\Vert u_1\Vert ^2=E(0)=d. \end{aligned}$$

The remainder proof is similar to part (i) of this Theorem. $\square$

Lemma 4.2

(Invariance of $V'$ for $E(0)=d$, Lemma 2.7 in [22]) Let f(u) satisfy (H), $u_0(x)\in H^1({\mathbb {R}}^n)$, $u_1(x)\in L^2({\mathbb {R}}^n)$. Suppose that $E(0)=d$ and $(u_0,u_1)\ge 0$, then the set

$$\begin{aligned} V'=\left\{ u\in H^1({\mathbb {R}}^n)\big |\ I(u)<0\right\} \end{aligned}$$

is invariant under the flow of problem (1.1)–(1.3).

Theorem 4.3

(Finite time blow up for $E(0)=d$) Let f(u) satisfy (H), $u_0(x)\in H^1({\mathbb {R}}^n)$, $u_1(x)\in L^2({\mathbb {R}}^n)$. Assume that $E(0)=d$, $I(u_0)< 0$ and $(u_0,u_1)\ge 0$. Then the solution to problem (1.1)–(1.3) blows up in finite time.

Proof

Let u(t) be any weak solution to problem (1.1)–(1.3) with $E(0)=d$, $I(u_0)< 0$ and $(u_{0},u_{1})\ge 0$ and T be the maximum existence time of u(t). We prove $T<+\infty$. Arguing by contradiction, we suppose $T=\infty$. Recalling auxiliary function M(t) as (3.7) shows and from Lemma 4.2, we have

$$\begin{aligned} M''(t)=2\Vert u_t\Vert ^2+2\langle u_{tt},u\rangle =2\Vert u_t\Vert ^2-2I(u)>0,\ t\in (0,+\infty ), \end{aligned}$$

(4.5)

which implies that $M'(t)$ is strictly increasing on $(0,\infty ).$ Hence for any $t_{0}>0$, we get

$$\begin{aligned} M'(t)> M'(t_0)> M'(0)\ge 0,\ t\in (0,+\infty ), \end{aligned}$$

then

$$\begin{aligned} M(t)\ge M'(t_0)(t-t_0)+M(t_0)\ge M'(t_0)(t-t_0),\ t\in (0,+\infty ). \end{aligned}$$

Similarly arguments to Theorem 3.2, we derive the conclusion. $\square$

5 Finite time blow up for $E(0)>0$

Lemma 5.1

Let $u_0(x)\in H^1({\mathbb {R}}^n)$, $u_1\in L^2({\mathbb {R}}^n)$ and $(u_{0},u_{1}) \ge 0$. Suppose that u is a solution of the problem (1.1)–(1.3), then the map $\{t\mapsto \Vert u\Vert ^2\}$ is strictly increasing as long as $u(t)\in V'$.

Proof

Recalling the function M(t) as (3.7) shows and by (3.9) and $u\in V'$, we have

$$\begin{aligned} M''(t)=2\langle u,u_{tt}\rangle +2\Vert u_t\Vert ^2=-2I(u)+2\Vert u_t\Vert ^{2}>0,\ t\in (0,+\infty ). \end{aligned}$$

Similarly arguments to Theorem 4.3, we know that M(t) is strictly increasing on $[0,+\infty )$. $\square$

Lemma 5.2

(Invariance of the unstable set $V'$ for $E(0)>0$) Let $u_0(x)\in H^{1}({\mathbb {R}}^n)$ and $u_1(x)\in L^{2}({\mathbb {R}}^n)$. Assume that (H), $(u_{0},u_{1})\ge 0$, $u_0\in V'$ and

$$\begin{aligned} \Vert u_0\Vert ^2>\frac{2(p+1)}{p-1}E(0)>0 \end{aligned}$$

(5.1)

hold, then $u\in V'$ for all $t\in [0,T)$.

Proof

We prove $u(t)\in V'$ for all $t\in [0,T)$. By contradiction, suppose that there is a $t_0\in (0,T)$ such that $u\in {\mathcal {N}}$ and $I(u(t))<0$ for all $t\in [0,t_0)$. The Lemma 5.1 tells that M(t) is strictly increasing on the interval $[0,t_0)$, which implies that

$$\begin{aligned} M(t)>\Vert u_0\Vert ^2>\frac{2(p+1)}{p-1}E(0),\ t\in [0,t_0). \end{aligned}$$

Thus the continuity of u(t) in time tells

$$\begin{aligned} M(t_0)=\Vert u(t_0)\Vert ^2>\frac{2(p+1)}{p-1}E(0). \end{aligned}$$

(5.2)

Then by (2.2) and (2.5), for $t\in [0,t_0]$ we obtain

$$\begin{aligned} E(0)=E(t)=&\frac{1}{2}\Vert u_t\Vert ^2+\frac{1}{2}\Vert u\Vert _{H^1}^2-\sum _{k=1}^{l}\frac{a_{k}}{p_{k}+1}\Vert u\Vert _{p_{k}+1}^{p_{k}+1}+ \sum _{j=1}^{s}\frac{b_{j}}{q_{j}+1}\Vert u\Vert _{q_{k}+1}^{q_{k}+1}\nonumber \\ \ge&\frac{1}{2}\Vert u_t\Vert ^2+\frac{p-1}{2(p+1)}\Vert u\Vert ^2+\frac{1}{p+1}I\left( u\right) . \end{aligned}$$

(5.3)

We substitute $t=t_{0}$ into (5.3) and by the fact that $I(u(t_{0}))=0$ to obtain

$$\begin{aligned} \Vert u(t_0)\Vert ^2\le \frac{2(p+1)}{p-1}E(0), \end{aligned}$$

which contradicts (5.2). So we complete the proof. $\square$

Theorem 5.3

(Finite time blow up for $E(0)>0$) Let f(u) satisfy (H), $u_0(x)\in H^{1}({\mathbb {R}}^{n})$ and $u_1(x)\in L^{2}({\mathbb {R}}^n)$. Suppose that $E(0)>0$, $I(u_0)<0$, $(u_{0},u_{1})\ge 0$ and (5.1) hold, then the corresponding solution u(x, t) of problem (1.1)–(1.3) blows up in finite time.

Proof

By contradiction, we suppose that u(t) is global in time. For any $T> 0$, from (3.7), the Schwarz inequality and (4.5), we obtain

$$\begin{aligned}&M''(t)M(t)-\frac{p+3}{4}(M'(t))^2\nonumber \\&\quad \ge M(t)\left( M''(t)-(p+3)\Vert u_t\Vert ^2\right) \nonumber \\&\quad = M(t)\left( -2I(u)-(p+1)\Vert u_t\Vert ^2\right) \\&\quad =M(t)\xi (t), \ t\in [0,T),\nonumber \end{aligned}$$

(5.4)

where

$$\begin{aligned} \xi (t):=-2I(u)-(p+1)\Vert u_t\Vert ^2,\ t\in [0,T). \end{aligned}$$

(5.5)

By (5.3), we have

$$\begin{aligned} (p+1)\Vert u_t\Vert ^2\le -(p-1)\Vert u\Vert ^2_{H^1}+2(p+1)E(0)-2I(u),\ t\in [0,T). \end{aligned}$$

(5.6)

We substitute (5.6) into (5.5), by Lemma 5.1 and (5.1) to obtain

$$\begin{aligned} \xi (t)\ge (p-1)\Vert u_{0}\Vert ^{2}-2(p+1)E(0)>0,\ t\in [0,T), \end{aligned}$$

then

$$\begin{aligned} \xi (t)>\delta ,\ t\in [0,T) \end{aligned}$$

(5.7)

for a constant $\delta >0$. On the other hand, the Lemma 5.2 tells that $I(u(t))<0$ for all $t\in [0,T)$. Similar arguments in Lemma 5.1, we know that M(t) is strictly increasing on [0, T). The continuity of u(t) in t tells

$$\begin{aligned} M(t)\ge \rho ,\ t\in [0,T) \end{aligned}$$

for a constant $\rho >0$. Hence from (5.4) and (5.7), we have

$$\begin{aligned} M''(t)M(t)-\frac{p+3}{4}(M'(t))^2>\rho \delta ,\ t\in [0,T). \end{aligned}$$

Then similar arguments in the proof of Theorem 3.2, we achieve the conclusion. $\square$