Global existence and blow-up of solutions for logarithmic Klein-Gordon equation

1.   Introduction

In this paper, we consider the following problem

where $\Omega\subset R^n$ is a bounded domain with smooth boundary $\partial\Omega$.

The model equation (1.1) arises in logarithmic quantum mechanics and is applied to nuclear physics, optics and geophysics ^[1,2,3,4,5]. P. Gorka ^[6] dealt with the equation

with initial datum

and boundary value condition

where ${\mathcal{O}} = [a, b]\subset R^1$, $\varepsilon\in [0, 1]$. By applying the Galerkin method, logarithmic Sobolev inequality and compactness theorem, he established the global weak solutions of the problem (1.4)–(1.6). K. Bartkowski and P. Korka ^[7] showed the classical solutions and weak solutions to the Cauchy problem of Eq (1.4) for ${\mathcal{O}} = R^1$. In ^[8], T. Cazenave and A. Haraux investigated the local and global solutions for the Cauchy problem of the logarithmic wave equation $u_{tt}-\Delta u = u\log|u|.$

For the following nonlinear Klein-Gordon equation

For $n\geq3$, P. Brenner ^[9] studied $L_p-$decay and scattering properties for the Cauchy problem of the Eq (1.7). As $n = 1$ and 2, K. Nakanishi ^[10] showed that the scattering operators for Eq (1.7) are well-defined in whole energy space in $R^{1+n}$ with $p > 1+\frac4n$. Under the condition of small energy data, such results were known for $n\geq3$ ^[11,12,13].

As $m = 0$, for sufficiently large initial data, the blow-up results of the problem (1.7)–(1.9) in finite time was proved by H. A. Levine ^[14] and J. Ball ^[15]. Furthermore, Y. C. Liu ^[16], L. E. Payne and D. H. Sattinger ^[17] and D. H. Sattinger ^[18] obtained the results of the global existence and nonexistence of weak solutions for the problem (1.7)–(1.9) by establishing the method of potential wells. Also in ^[16,19], the authors gave a threshold result of solutions and obtained the vacuum isolating of solutions.

At last we should mention that the logarithmic heat equation was studied by H. Chen and S. Y. Tian ^[20] and H. Chen, P. Luo and G. W. Liu ^[21]. Moreover, there were also many researches on the logarithmic Schrödinger equation ^{[22,23,24,25]}.

In this paper, by applying Galerkin method and compactness criterion, we prove the global existence of the problem (1.1)–(1.3). Furthermore, in the sense of $L^2$ norm, the blow-up result for this problem is obtained by the concavity method.

2.   Preliminaries

2.1.   Some lemmas

For the applications through this paper, we list up some known lemmas.

Definition 2.1 If

satisfies

Then the function $u$ is called a weak solution of (1.1)–(1.3) on $[0, T]$.

Lemma 2.1 Assume that $2\leq r < +\infty, \ n\leq2$ and $2\leq r\leq\frac{2n}{n-2}, \ n > 2$. Then

where $C > 0$ is a constant depending on $\Omega$ and $r$.

Lemma 2.2 (^[20,21,26]) If $u\in H_0^1(\Omega)$, then for each $a > 0$, one has the inequality

Lemma 2.3 Let $u(t)$ be a solution of the problem (1.1)–(1.3), then the energy ${\mathcal{E}}(t)$ is conservation. Namely, ${\mathcal{E}}(t) = {\mathcal{E}}(0), \ \forall t > 0$, where

for $u\in H_0^{1}(\Omega), \ t\geq0$ and

is the initial total energy.

Lemma 2.4^[27] Let $X$ be a Banach space, if $f\in L^p(0, T; X), \ \frac{\partial f}{\partial t}\in L^p(0, T; X)$, then $f$ is a continuous injection from $[0, T]$ on to $X$ when the value is transformed in the set of measure zero in $[0, T]$.

Lemma 2.5^[28] Let $u_n(x)$ be a bounded sequence in $L^p(\Omega), \ 1\leq p < +\infty$ such that $u_n$ almost everywhere converges to $u$. Then $u\in L^p(\Omega)$ and $u_n$ weakly converges in $L^p(\Omega)$ to $u$, where $\Omega\subset R^n$ is a bounded domain.

The local existence result of the problem (1.1)–(1.3) is described as follows. For its detailed proof process, see references ^[31,32,33].

Theorem 2.1 (Local existence) Let $u_0\in H_0^{1}(\Omega), \ u_1\in L^2(\Omega)$. Then there exists $T > 0$ such that the problem (1.1)–(1.3) has a unique local solution $u(t)$ satisfying

2.2.   Potential wells

At first, we introduce some useful functionals

By (2.1), (2.3) and (2.4), we have

for $u\in H_0^{1}(\Omega)$.

As in ^[17], the potential well depth is defined as

Now, we define the Nehari manifold (^[29,30]) by

The stable set $\mathcal{W}$ and the unstable set $\mathcal{U}$ can be defined respectively by

and

It is to see that the potential well depth $d$ may also be described as

Lemma 2.6 Let $u\in H_0^{1}(\Omega)$ and $\|u\|\not = 0$, then we have

where

Proof. (ⅰ) By $\lim_{\lambda\rightarrow0^+}\lambda^2\log\lambda = 0, \ \ \lim_{\lambda\rightarrow +\infty}\log \lambda = +\infty$ and

we get

(ⅱ) By direct calculations, we obtain

Let $\frac d{d\lambda}{\mathcal{J}}(\lambda u) = 0$, then we deduce that

From (3.2), we have

By (2.9) and (2.10), the equality (2.8) is valid.

Lemma 2.7 If $u\in H_0^{1}(\Omega)$, then

Proof. By Lemma 2.2, we have

By taking $a = \sqrt{2\pi}$, we obtain from (2.12) that

Combining Lemma 2.6 and (2.5) yields that

We receive from (2.13) and Lemma 2.6 that

then

It follows from(2.14) and (2.15) that

By (2.6) and (3.16), we have that $d = \frac14(\sqrt{2\pi})^ne^{n+2}.$

In order to further study the problem (1.1)–(1.3), for $0 < \varepsilon < 1$ and $u\in H_0^{1}(\Omega)$, we define some functionals as follows

Let

then we define $d(\varepsilon)$ as

Proposition 2.1 If $d(\varepsilon)$ is defined by (2.20), then

where $\lambda_1$ is the first eigenvalue of the following boundary value problem

Proof. By $u\in {\mathcal{N}}_\varepsilon(u)$, we get

From (2.17) and Lemma 2.2 that

By taking $a^2 = 2\pi\varepsilon$ in (2.24), we obtain

which implies that

Since the eigenvalue $\lambda_1$ satisfies the problem (2.22), so we gain

It follows from (2.23) and (2.25) that

Thus, by (2.20), we have

Proposition 2.2 As a function of $\varepsilon$, $d(\varepsilon)$ have the following properties for $\varepsilon\in[0, 1]$:

(a) $d(0) = d(1) = 0.$

(b) $d(\varepsilon)$ is increasing on $[0, \varepsilon_0]$ and decreasing on $[\varepsilon_0, 1]$. Thus, $d(\varepsilon)$ gets the maximum at $\varepsilon_0 = \frac n{n+2}$, and $d(\varepsilon_0) = \frac{4\lambda_1de}{n+2}(\frac n{n+2})^{\frac n2}.$

(c) For $\forall h\in(0, d(\varepsilon_0))$, the equation $d(\varepsilon) = h$ has two roots $\varepsilon_1$ and $\varepsilon_2$ in the interval $(0, \varepsilon_0)$ and $(\varepsilon_0, 1)$, respectively.

Proof. (a) is easy to be proved. Here we omit the proof of it.

(b) By calculation, we have

From $d'(\varepsilon) = 0$, we conclude that $\varepsilon_0 = \frac n{n+2}.$ In addition, by (2.26), we get $d'(\varepsilon) > 0$ on $(0, \varepsilon_0)$ and $d'(\varepsilon) < 0$ on $(\varepsilon_0, 1)$. Therefore, $d(\varepsilon)$ takes the maximum value at $\varepsilon_0 = \frac n{n+2}$, and

(c) Let $f(\varepsilon) = d(\varepsilon)-h$, then $f(0) = -h < 0, \ f(\varepsilon_0) = d(\varepsilon_0)-h > 0, \ f(1) = -h < 0$. According to the continuity of function $f(\varepsilon)$ on interval $[0, 1]$, the equation $f(\varepsilon) = 0$ i.e. $d(\varepsilon) = h$ has two roots $\varepsilon_1\in(0, \delta_0)$ and $\varepsilon_2\in(\delta_0, 1)$.

Proposition 2.3 Let $r(\varepsilon) = 4\lambda_1de\varepsilon^{\frac n2},$ then

(1) If ${\mathcal{J}}(u)\leq d(\varepsilon)$, then $0 < \|\nabla u\|^2\leq r(\varepsilon)$ if only and if $J_\varepsilon(u)\geq0.$

(2) If ${\mathcal{J}}_\varepsilon(u) < 0$, then $\|\nabla u\|^2 > r(\varepsilon)$.

Proof. (1) For $a^2 = 2\pi\varepsilon$, we have

By $0 < \|\nabla u\|^2\leq r(\varepsilon)$, we see that $\log\frac{4\lambda_1de\varepsilon^{\frac n2}}{\|\nabla u\|^2}\geq0$. Therefore, we conclude from (2.27) that ${\mathcal{J}}_\varepsilon(u)\geq0$.

If ${\mathcal{J}}_\varepsilon(u)\geq0$, then from (2.20) and

we have

which implies that $\|\nabla u\|^2\leq r(\varepsilon).$

(b) By (2.24) and ${\mathcal{J}}_\varepsilon(u) < 0$, we have $\log\frac{4\lambda_1de\varepsilon^{\frac n2}}{\|\nabla u\|^2} < 0$, which implies that $\|\nabla u\|^2 > r(\varepsilon).$

On the basis of Proposition 2.1 and Proposition 2.2, we define a family of potential wells by

and

for $\varepsilon\in(0, 1)$.

Remark From ${\mathcal{J}}_\varepsilon(u) > 0$ and

we have ${\mathcal{J}}(u) > 0.$

3.   Global existence of solutions

In this section, by applying Galerkin method and the compactness principle, we study the global solutions of the problem (1.1)–(1.3).

Theorem 3.1 If $u_0\in {\mathcal{W}}, \ u_1\in L^2(\Omega)$ satisfy $0 < {\mathcal{E}}(0) < d$, then the problem (1.1)–(1.3) admits a global solution $u(x, t)$ such that $u(x, t)\in L^\infty([0, +\infty); H_0^{1}(\Omega)), \ u_t(x, t)\in L^\infty([0, +\infty); L^2(\Omega))$.

Proof. Let $\{\omega_j\}_{j = 1}^{\infty}$ be a basis for $H_0^{1}(\Omega)$. We are going to find out the approximate solution $u_m(t)$ in the form $u_m(t) = \sum\limits_{j = 1}^{m}g_{jm}(t)\omega_j$ with $g_{jm}(t)\in C^2[0, T], \ \forall T > 0$, where the unknown functions $g_{jm}(t)$ are determined by the following ordinary differential equation

with initial data

By the density of $H_0^{1}(\Omega)$ in $L^2(\Omega)$, there exist $\alpha_{jm}$ and $\beta_{jm}, j = 1, 2, \cdots, m$ such that

By a Picard's iteration method, there exists solution $g_{jm}(t)$ of the problem (3.1) and (3.2) in interval $[0, t_m^1)$ for some $t_m^1\leq T$. From the uniformly boundedness of function $g_{jm}(t)$ and the extension theorem, we can extend this solution to the whole interval $[0, T]$ for any given $T > 0$ by making use of the a priori estimates below.

Multiplying both sides of (3.1) by $g_{jm}'(t)$ and summing with respect to $j$ from $1$ to $m$, and integrating over $[0, t]$, we have from (2.1) and (2.3) that

By (3.5), we can verify

In fact, suppose that (3.6) is false and let $\tau$ be the smallest time for that $u_m(\tau)\notin \mathcal{W}$. Then in virtue of the continuity of $u_m(t)$, we see $u_m(\tau)\in\partial \mathcal{W}$. From the continuity ${\mathcal{J}}(u(t))$ and ${\mathcal{K}}(u(t))$ with respect to $t$, we have either ${\mathcal{J}}(u_m(\tau)) = d$ or ${\mathcal{K}}(u_m(\tau)) = 0$. By (3.5), we get ${\mathcal{J}}(u_m(\tau)) < d.$ So, the former case is impossible. Assume that ${\mathcal{K}}(u_m(\tau)) = 0$ is valid, then $u_m(\tau)\in\mathcal{ N}$. From (2.7), we obtain ${\mathcal{J}}(u_m(\tau))\geq d$ which is contradictive with (3.5). Therefore, the latter case is impossible as well.

We deduce from (2.5), (3.5) and (3.6) that

which implies that

From (2.1), (3.5) and Lemma 2.2, we obtain

Let $a = \sqrt{\pi}$, then we have from (3.8) and (3.9)

which implies

We know that $u_{mtt}(t)$ is uniformly bounded in $L^\infty(0, T; H^{-1}(\Omega))$ by a standard discussion. Then, there exists a function $u(t)$ and a convergent subsequence of $\{u_\mu\}$, still denoted by $\{u_m\}$. As $m\rightarrow \infty$, we obtain

From (3.12)–(3.14) and Aubin-Lions lemma, we have

which implies

By (3.16), we can infer that

Let $\Omega_1 = \{x\in\Omega; \ |u_m(x)|\leq1\}$ and $\Omega_2 = \{x\in\Omega; \ |u_m(x)| > 1\}$, then by direct calculation, we get from (3.11)

The estimate (3.18) indicates that $u_m\log|u_m|$ is uniformly bounded in $L^\infty(0, T; L^2(\Omega))$. Thus there exists a function $\chi$ such that

From (3.17), (3.18) and Lemma 2.5, we have

It follows from (3.19) and (3.20) that

Let $m\rightarrow \infty$ in (3.1), by using (3.12), (3.14), (3.19) and (3.20), we obtain

By the density of the system $\{\omega_j\}_{j = 1}^\infty$ in $H_0^1(\Omega)$, we deduce that

for $\forall \varphi\in H_0^1(\Omega)$. That is to say $u$ satisfies the Eq (1.1) in the weak sense.

Next, we prove that $u(0) = u_0, u_t(0) = u_1$ are held. It follows from (3.12), (3.13) and Lemma 2.4 that $u(t): [0, T]\rightarrow L^2(\Omega)$ is continuous. Hence, we gain that $u(0)$ is valid and $u_m(0)\rightarrow u(0)$ weakly in $L^2(\Omega)$. By (3.3), we obtain $u(0) = u_0$.

To prove $u_t(0) = u_1$, we note that

where $\xi(t)$ is a smooth function with $\xi(0) = 1, \xi(T) = 0$.

For given $j$, as $m\rightarrow \infty$, in the distribution sense, we have

in ${\mathcal{D'}}([0, T])$. On the other hand, by (3.1), we get

Taking the limitation on both sides of (3.23) as $m\rightarrow \infty$, we obtain

Therefore,

It follows from (3.22) and (3.24) that $(u_t(0), \omega_j) = (u_1, \omega_j)$. By the density of $\{\omega_j\}_{j = 1}^m$ in $L^2(\Omega)$, we get $u_t(0) = u_1$.

The proof of Theorem 3.1 is completed.

For the case ${\mathcal{K}}(u_0)\geq0$ and ${\mathcal{E}}(0) = d$, the global existence result of the problem (1.1)–(1.3) reads as follows:

Theorem 3.2 Given that $u_0\in H_0^{1}(\Omega), \ u_1\in L^2(\Omega)$. If ${\mathcal{E}}(0) = d$ and ${\mathcal{K}}(u_0)\geq0$, then there exists a global weak solution $u(x, t)$ for the problem (1.1)–(1.3) such that $u(x, t)\in L^\infty([0, +\infty); H_0^{1}(\Omega)), \ \ u_t(x, t)\in L^\infty([0, +\infty); L^2(\Omega)).$

Proof. Let $\rho_k = 1-\frac1k$ and $u_{0k} = \rho_k u_0$ for $k\geq2$. We consider the following problem

By $K(u_0)\geq0$ and Lemma 2.6, we have $\lambda^* = \lambda^*(u_0)\geq1$. Therefore, we conclude that ${\mathcal{K}}(u_{0k}) > 0$. Thus, we have

and ${\mathcal{J}}(u_{0k}) = {\mathcal{J}}(\mu_ku_0) < {\mathcal{J}}(u_0)$. Therefore,

So, we obtain $u_{0k}\in \mathcal{W}$. For each $k$, by Theorem 3.1, the problem (3.25) admits a global weak solution $u_k(t)$ which satisfies that $u_k(t)\in L^\infty([0, +\infty); H_0^{1}(\Omega)), \ \ u_{kt}(t)\in L^\infty([0, +\infty); L^2(\Omega))$ and

for any $\varphi\in H_0^{1}(\Omega)$. In addition,

By using the formula (3.27) and the same argument as (3.6), we may verify $u_k(t)\in \mathcal{W}.$

The remainder proof for Theorem 3.2 is the same process as Theorem 3.1. Here, we omit it.

Next, we study the global existence of solution to the problem (1.1)–(1.3) in a family of potential wells $\mathcal{W}_\varepsilon$. For this purpose, we need the following lemmas

Lemma 3.1 Suppose that $u_0\in H_0^{1}(\Omega), \ u_1\in L^2(\Omega)$ and $0 < {\mathcal{E}}(0) < d(\varepsilon_0)$. $\varepsilon_1, \varepsilon_2$ are the two roots of the equation $d(\varepsilon) = {\mathcal{E}}(0)$, $u(x, t)$ is a solution of the problem (1.1)–(1.3). Then

(ⅰ) If ${\mathcal{J}}_{\varepsilon_0}(u_0) > 0$, then $u(t)\in \mathcal{W}_\varepsilon$ for $\forall \varepsilon\in(\varepsilon_1, \varepsilon_2)$.

(ⅱ) If ${\mathcal{J}}_{\varepsilon_0}(u_0) < 0$, then $u(t)\in \mathcal{U}_\varepsilon$ for $\forall \varepsilon\in(\varepsilon_1, \varepsilon_2)$.

Proof. Firstly, under the conditions in Lemma 4.1, we prove the sign of ${\mathcal{J}}_\varepsilon(u)$ is invariant on the interval $(\varepsilon_1, \varepsilon_2)$.

Multiplying both sides of the Eq (1.1) by $u_t$, then we get from integrating over $\Omega\times[0, t]$ that

By (3.28) and $0 < {\mathcal{E}}(0) < d(\varepsilon_0)$, it is easy to see that $0 < {\mathcal{J}}(u) < d(\varepsilon_0)$. Namely, $\|\nabla u\|\not = 0$.

By contradiction, we suppose that the sign of ${\mathcal{J}}_\varepsilon(u)$ is variable on $(\varepsilon_1, \varepsilon_2)$, then there exists $\varepsilon'\in(\varepsilon_1, \varepsilon_2)$ such that ${\mathcal{J}}_{\varepsilon'}(u) = 0$. From (3.28), (2.20) and Proposition 2.2, we gain that ${\mathcal{E}}(0)\geq {\mathcal{J}}(u)\geq d(\varepsilon') > d(\varepsilon_1) = d(\varepsilon_2),$ which is contradictive with ${\mathcal{E}}(0) = d(\varepsilon_1) = d(\varepsilon_2)$.

(ⅰ) Because ${\mathcal{J}}_{\varepsilon_0}(u_0) > 0$ and the sign of ${\mathcal{J}}_\varepsilon(u)$ is not changed for $(\varepsilon_1, \varepsilon_2)$, we have $\|\nabla u_0\|\not = 0$ and ${\mathcal{J}}_{\varepsilon}(u_0) > 0, \ \forall\varepsilon\in(\varepsilon_1, \varepsilon_2)$. From (3.28), we get ${\mathcal{J}}(u_0)\leq {\mathcal{E}}(0) < d(\varepsilon)$. Thus, we obtain $u_0\in {\mathcal{W}}_\varepsilon, \forall \varepsilon\in(\varepsilon_1, \varepsilon_2)$.

Next we prove $u(t)\in {\mathcal{W}}_\varepsilon$ for $\forall\varepsilon\in(\varepsilon_1, \varepsilon_2)$ and $0 < t < T$, where $T$ is the existence time of $u(t)$. Assume that there exists a number $t_1\in (0, T)$ such that $u(t_1)\not\in {\mathcal{W}}_\varepsilon$. Then, in virtue of the continuity of $u(t)$, we see $u(t_1)\in \partial {\mathcal{W}}_\varepsilon, \forall \varepsilon\in(\varepsilon_1, \varepsilon_2)$. From the definition of ${\mathcal{W}}_\varepsilon$ and the continuity of ${\mathcal{J}}(u(t))$ and ${\mathcal{J}}_\varepsilon(u(t))$ with respect to $t$, we have

or

It follows from (3.28) that

Thus, the case (3.30) is impossible. If (3.29) holds, then, by (3.17), we have ${\mathcal{J}}(u(t_1))\geq d(\varepsilon)$ which is contradictive with (3.31). Consequently, the case (3.29) is also impossible. Thus, we conclude that $u(t)\in {\mathcal{W}}_\varepsilon, \ \forall\varepsilon\in(\varepsilon_1, \varepsilon_2)$.

(ⅱ) Since the sign of ${\mathcal{J}}_\varepsilon(u)$ is not changed for $(\varepsilon_1, \varepsilon_2)$, by ${\mathcal{J}}_{\varepsilon_0}(u_0) < 0$, we get ${\mathcal{J}}_{\varepsilon}(u_0) < 0$ for $\forall \varepsilon\in(\varepsilon_1, \varepsilon_2)$. Thus, we have $u_0\in {\mathcal{U}}_\varepsilon$ from ${\mathcal{J}}(u_0) < {\mathcal{E}}(0) = d(\varepsilon)$. Now we prove $u(t)\in {\mathcal{U}}_\varepsilon$ for $\forall\varepsilon\in(\varepsilon_1, \varepsilon_2), \ 0 < t < T$. If it is not right, then there exists $t_2\in(0, T)$ with $u(t_2)\in \partial {\mathcal{U}}_\varepsilon$ for $\forall \varepsilon\in(\varepsilon_1, \varepsilon_2)$, i.e. either ${\mathcal{J}}_{\varepsilon}(u(t_2)) = 0$ or ${\mathcal{J}}(u(t_2)) = d(\varepsilon)$. By (3.31), ${\mathcal{J}}(u(t_2)) = d(\varepsilon)$ is impossible. Moreover, let $t_2$ be the first time such that ${\mathcal{J}}_{\varepsilon}(u(t_2) = 0$, then ${\mathcal{J}}_{\varepsilon}(u(t)) < 0$ for $0\leq t < t_2$. Combining (3.28) and Proposition 2.3, we get $\|\nabla u(t)\| > r(\varepsilon)$ for $t\in[0, t_2)$. Hence, we obtain $\|\nabla u(t_2)\|\geq r(\varepsilon)$. From (3.17), it follows that ${\mathcal{J}}(u(t_2))\geq d(\varepsilon)$ which is contradictive with (3.31). This implies that ${\mathcal{J}}_{\varepsilon}(u(t_2)) = 0$ is also impossible. Therefore, we have $u(t)\in {\mathcal{U}}_\varepsilon, \ \forall\varepsilon\in(\varepsilon_1, \varepsilon_2)$.

Lemma 3.2 Assume that $u_0\in H_0^{1}(\Omega), \ u_1\in L^2(\Omega)$ and $0 < {\mathcal{E}}(0)\leq h < d(\varepsilon_0)$. $\varepsilon_1, \varepsilon_2$ are the two roots of the equation $d(\varepsilon) = h$, $u(x, t)$ are solutions of the problem (1.1)–(1.3). Then

(ⅰ) If ${\mathcal{J}}_{\varepsilon_0}(u_0) > 0$, then $u(t)\in {\mathcal{W}}_\varepsilon$ for $\forall \varepsilon\in(\varepsilon_1, \varepsilon_2)$.

(ⅱ) If ${\mathcal{J}}_{\varepsilon_0}(u_0) < 0$, then $u(t)\in U_\varepsilon$ for $\forall \varepsilon\in(\varepsilon_1, \varepsilon_2)$.

We can prove Lemma 3.2 by means of the similar method shown in the proof of Lemma 3.1. Here, we omit it.

Theorem 3.3 Suppose that $\varepsilon_1, \varepsilon_2$ are the two roots of the equation $d(\varepsilon) = {\mathcal{E}}(0)$ and ${\mathcal{J}}_{\varepsilon_2}(u_0) > 0$. If $(u_0, u_1)\in H_0^{1}(\Omega)\times L^2(\Omega)$ and $0 < {\mathcal{E}}(0) < d(\varepsilon)$, then the problem (1.1)–(1.3) admits a global weak solution $u(x, t)$ such that

for any $T > 0$.

Proof. By using the similar argument as Theorem 3.1, we are going to prove Theorem 3.3. Under the conditions in Theorem 3.3, by Lemma 3.1, we have $u_0\in {\mathcal{W}}_\varepsilon$ for $\varepsilon\in(\varepsilon_1, \varepsilon_2)$. For any given $\varepsilon_1 < \varepsilon < \varepsilon_2$, we derive ${\mathcal{J}}_\varepsilon(u_m(0)) > 0$ and ${\mathcal{E}}_m(0) < d(\varepsilon)$, which implies that $u_m(0)\in {\mathcal{W}}_\varepsilon$. Once again, we get $u_m(t)\in {\mathcal{W}}_\varepsilon$ by Lemma 3.1. Here, the approximate solutions $u_m(t)$ are given in the proof of Theorem 3.1.

Multiplying both sides of (3.1) by $g_{jm}'(t)$, summing over $j$ from $1$ to $m$ and integrating with respect to $t$, we obtain

From (2.3) and (2.17), we deduce

Combining (2.21), (3.32), (3.33), by $u_m(t)\in {\mathcal{W}}_\varepsilon$, we get ${\mathcal{J}}(u_m(t)) > 0$ and the following estimates

From (2.21) and (3.32), we find that

By means of the same procedure as the estimates (3.18), we obtain

The remainder of the proof for Theorem 3.3 is the same as those of Theorem 3.1. Here, we omit them.

4.   Blow-up of solution

In this section, we establish the blow-up property of solution for the problem (1.1)–(1.3).

Lemma 4.1 Let $u(t)$ be a solution of (1.1)–(1.3). If $u_0\in{\mathcal{U}}$ and ${\mathcal{E}}(0) < d$, then $u(t)\in{\mathcal{U}}$ and ${\mathcal{E}}(t) < d$, for all $t\geq0$.

Proof. It follows from Lemma 2.3 that

By (2.5), we obtain

By contradiction, we assume that there exists $t^*\in[0, +\infty)$ such that $u(t^*)\not\in{\mathcal{U}}$, then, from the continuity of ${\mathcal{K}}(u(t))$ on $t$, we have ${\mathcal{K}}(u(t^*)) = 0$. This implies that $u(t^*)\in{\mathcal{N}}$. We get from (2.7) that ${\mathcal{J}}(u(t^*))\geq d$, which is contradiction with (4.1). Consequently, Lemma 4.1 is valid.

Lemma 4.2 Suppose that $u\in {\mathcal{U}}$, then ${\mathcal{K}}(u(t)) < 2[{\mathcal{J}}(u(t))-d].$

Proof. If $u\in {\mathcal{U}}$, then it follows from Lemma 2.6 that there exists a $\lambda^*$ such that $0 < \lambda^* < 1$ and ${\mathcal{K}}(\lambda^*u) = 0$. By the definition of $d$ in (2.6), we get

We have from (2.5) that $d < {\mathcal{J}}(u(t))-\frac12{\mathcal{K}}(u(t)),$ which implies that ${\mathcal{K}}(u(t)) < 2[{\mathcal{J}}(u(t))-d].$

Theorem 4.1 If the initial datum $u_0\in{\mathcal{U}}, \ u_1\in L^2(\Omega)$ satisfy that ${\mathcal{E}}(0) < d$ and $\int_\Omega u_0u_1dx > 0$, then the solution $u(t)$ in Theorem 2.1 of the problem (1.1)–(1.3) blows up as time $t$ goes to infinity, which means that

Proof. Let ${\mathcal{P}}(t) = \|u(t)\|^2$, then ${\mathcal{P}}(t) > 0, \ t\geq0$. Direct computations show that

From (1.1) and (2.4), we get

It follows from Cauchy-Schwarz inequality and (4.2) that

Then we have from (2.5) and Lemma 2.3 that

From $u_0\in{\mathcal{U}}, \ {\mathcal{E}}(0) < d$ and Lemma 4.1, we have $u\in{\mathcal{U}}, \ {\mathcal{E}}(t) < d$. Hence by Lemma 4.2, we obtain that

We conclude from (4.5) and (4.6) that

Furthermore, by direct calculation, it is easy to see that

We know from (4.9) that the function $(\log|{\mathcal{P}}(t)|)' = \frac{{\mathcal{P}}'(t)}{{\mathcal{P}}(t)}$ is increasing on time $t$. By integrating both sides of (4.8) from $0$ to $t$, we get

for $t > 0.$ Therefore,

From the definition of ${\mathcal{P}}(t)$, (4.10) means that

This finishes the proof of Theorem 4.1.

5.   Conclusions

By applying logarithmic Sobolev inequality, the Galerkin method and compactness theorem, we prove the global existence results of the problem (1.1)–(1.3) under the conditions that the initial values $u_0\in {\mathcal{W}}, \ u_1\in L^2(\Omega)$ satisfy (ⅰ) $0 < {\mathcal{E}}(0) < d$ or (ⅱ) ${\mathcal{K}}(u_0)\geq0$ and ${\mathcal{E}}(0) = d$. Meanwhile, under the condition of positive initial energy, by using the concavity analysis method, we establish the finite time blow-up result of solutions in the sense of $L^2$ norm. On the other hand, the global existence of solution for this problem is also obtained in a family of potential wells $\mathcal{W}_\varepsilon$. Our result implies that the polynomial nonlinearity is important for the solutions of such kinds of Klein-Gordon equation to be blow-up in finite time.

Acknowledgments

The authors would like to thank the reviewers and editors for their help to improve the quality of this article. Moreover, this Research was supported by Natural Science Foundation of Zhejiang Province (No. LY17A010009).

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.