Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy

1.   Introduction

The aim of this paper is to study the global behavior of the solutions to the Cauchy problem for the nonlinear Klein-Gordon equation

with critical initial energy $E(0) = d$. The nonlinear term $f(u)$ has one of the following forms

where the constants $a_k$, $p_k$ $(k = 1, 2, \ldots, l)$ and $b_j$, $q_j$ $(j = 1, 2, \ldots, s)$ fulfill the conditions

The combined power type nonlinearity (2) appears in numerous models of quantum mechanics, field theory, nonlinear optics and others. For example, the quadratic-cubic nonlinearity $f(u) = u^2+u^3$ describes the dislocation of crystals, see [16], while the the cubic-quintic nonlinearity $f(u) = u^3+u^5$ arises in particles physics, see e.g. [21,14].

The global existence or finite time blow up of the solutions to (1) - (3) is fully investigated for nonpositive energy $E(0)\leq 0$ and for subcritical energy $0<E(0) <d$ by means of the potential well method. Here $d$ is the critical energy constant, defined in (8). Potential well method is suggested in [20] for the wave equation and further on is applied for wide class of nonlinear dispersive equations, e.g. for nonlinear Klein-Gordon equations see [1,18,19,24,30]. Within this method the sign of the Nehari functional $I(0)$, see (7), is crucial for the global behavior of the solutions to (1) - (3). More precisely, for $0<E(0)<d$ the solutions blow up for a finite time if $I(0)<0$ and they are globally defined if $I(0)\geq 0$.

The case of critical initial energy, i.e. $E(0) = d$, is treated in [5,9,26,28] for the wave and damped wave equations in bounded domains and for the Klein-Gordon equation – in [8,16,19,24]. In the above papers the global existence is proved under conditions $I(0)>0$ or $\|u(t, \cdot)\|_{\mathrm{H}^1(\mathbb{R}^n)} = 0$ without any restrictions on the sign of the scalar product $(u_0, u_1)$ of the initial data. In the same papers the finite time blow up is obtained when $I(0)<0$ and $(u_0, u_1)\geq 0$. The case $E(0) = d$, $I(0)<0$ and $(u_0, u_1)<0$ is investigated only in [9,16]. The asymptotic behavior of the global solutions to the wave equation in bounded domains is studied in [9], while for Klein-Gordon equation similar results are given in [16].

In the case of supercritical initial energy, i.e. $E(0)>d$, only partial results for global behavior of the solutions to (1) - (3) are reported in the literature. There are a few sufficient conditions on the initial data $u_0$ and $u_1$, which guarantee finite time blow up, see [3,4,8,11,12,13,19,22,23,27,29]. In these sufficient conditions the nonnegative sign of $(u_0, u_1)$ is crucial.

In our previous paper [4] we prove a necessary and sufficient condition for finite time blow up of the solutions to (1) - (3) for arbitrary positive initial energy $E(0)>0$. More precisely, if $u(t, x)$ is a solution to (1) - (3) defined in the maximal existence time interval $[0, T_m)$, $0<T_m\leq\infty$, then $u(t, x)$ blows up for a finite time $T_m<\infty$ if and only if there exists $b\in[0, T_m)$ such that

Let us emphasize once again that the sign condition $(u_0, u_1)\geq 0$ plays very important role in all known sufficient conditions for finite time blow up of the solutions to (1) - (3) with supercritical energy, as well as in the necessary and sufficient condition (4).

In the present paper we focus on the global behavior of the solutions to (1) - (3) with critical initial energy $E(0) = d$. We give new necessary and sufficient conditions for finite time blow up and global existence, which are based on the study of the qualitative properties to a new ordinary differential equation. This approach improves the concavity method of Levine. As a consequence of the necessary and sufficient conditions for finite time blow up, we get new, more general sufficient conditions on the initial data for finite time blow up. In the case $I(0)<0$, new necessary conditions on the initial data for global existence are proved. The asymptotic behavior of the global solutions with $I(0)<0$ is studied in a similar way as in [9], where the wave equation in bounded domains is considered.

The paper is organized in the following way. In Section 2 some preliminary results are given. Section 3 deals with the global behavior of the solutions to a new ordinary differential equation. The results are an improvement of the concavity method of Levine and allow us to formulate necessary and sufficient conditions for finite time blow up. The main results of the paper are formulated and proved in Section 4 and Section 5. In Section 4 the finite time blow up is treated, while Section 5 deals with the global existence of the solutions and their asymptotic behavior.

2.   Preliminary

We will use the following short notations for the functions $u(t, x)$ and $v(t, x)$ depending on $t$ and $x$

We have the following local existence result to the Cauchy problem (1) - (3), see e.g. [2,6,7].

Theorem 2.1. Problem (1) - (3) admits a unique local weak solution

in the maximal existence time interval $[0, T_m)$. Moreover,

(ⅰ)

(ⅱ) for every $t\in[0, T_m)$ the solution $u(t, x)$ satisfies the conservation law

where the energy functional $E(t)$ is defined by

Definition 2.2. The solution $u(t, x)$ to (1) - (3), defined in the maximal existence time interval $[0, T_m)$, $T_m\leq\infty$, blows up at $T_m$ if

In order to prove a necessary and sufficient condition for finite time blow up of the solutions to (1) - (3), we use the following equivalence between the blow up of the $\mathrm{H}^1$ and $\mathrm{L}^2$ norms of $u(t, x)$.

Lemma 2.3. Suppose $u(t, x)$ is the solution to (1) - (3) with $p_l<(n+4)/n$ in the maximal existence time interval $[0, T_m)$, $0<T_m\leq\infty$. Then the blow up of $\mathrm{H}^1$ norm of $u(t, x)$ is equivalent to the blow up of the $\mathrm{L}^2$ norm of $u(t, x)$ at $T_m$, i.e.

The proof of Lemma 2.3 is based on the Gagliardo - Nirenberg inequality. In one-dimensional case it is given in [4]. The multidimensional case is treated in a similar way and we omit the proof.

Let us recall some important functionals - the Nehari functional $I(u(t, \cdot))$ and the potential energy functional $J(u(t, \cdot))$, as well as the critical energy constant $d$. When $u$ depends on $x$ and $t$ we use the short notations $I(u(t, \cdot)) = I(t)$ and $J(u(t, \cdot)) = J(t)$, i.e.

In the framework of the potential well method there are two important subsets of $\mathrm{H}^1(\mathbb{R}^n)$:

In the following theorem we formulate the sign preserving properties of $I(u)$, i.e. the invariance of $V$ and $W$ under the flow of (1) - (3) when $E(0) = d$.

Theorem 2.4. Suppose $u(t, x)$ is the weak solution of (1) - (3) defined in the maximal existence time interval $[0, T_m)$ and $E(0) = d$.

(ⅰ) If $u_0 \in W$, then $u(t, x) \in W$ for every $t \in[0, T_m)$;

(ⅱ) If $u_0 \in V$, then $u(t, x) \in V$ for every $t \in[0, T_m)$.

Proof. (ⅰ) Suppose $u_0 \in W$ but the result in (ⅰ) fails. Then for some $t_0\in (0, T_m)$ we have $u(t, x) \in W$ for $t \in [0, t_0)$ and $u(t_0, x) \in \partial W$, i.e. $I(t)>0$ or $\|u\|_1 = 0$ for $t \in [0, t_0)$ but $I(t_0) = 0$ and $\|u(t_0, \cdot)\|_1\ne 0$. Hence $u(t_0, \cdot)\in \mathcal{N}$ and from (5), (6), (8) it follows that $J(t_0) \geq d$ and the following inequalities hold

Hence

If $\hat{u}(x)$ is a ground state solution of (1), then $\hat{u}(x)$ satisfies the equation

Consequently, condition (9) means that the function $u(t_0, x)$ coincides with some ground state solution of (1). Without loss of generality we assume that $u(t_0, x) = \hat{u}(x)$. Since $\frac{\partial}{\partial t}\hat{u}(x) = \frac{\partial}{\partial t} u(t_0, x) = 0$, from the uniqueness of the weak solution to (1) - (3) we get $u(t, x) = u(t_0, x) = \hat{u}(x)$ for every $t \in [0, T_m)$, $x \in \mathbb{R}^n$. Hence from (9) it follows that $I(t) = I(t_0) = 0$ for every $t \in [0, T_m)$ and for $t = 0$ we get $I(0) = 0$, $\|u_0\|_1 = \|\hat{u}\|_1\ne 0$, which contradicts the assumption $u_0 \in W$. Thus statement (ⅰ) in Theorem 2.4 is proved.

(ⅱ) Suppose $u_0 \in V$. If $u(t_0, x) \not\in V$ for some $t_0 \in (0, T_m)$, then either $u(t_0, x) \in W$ or $I(t_0) = 0$ and $\|u(t_0, \cdot)\|_1\ne 0$. If $u(t_0, x) \in W$, then from (ⅰ) it follows that $u(t, x) \in W$ for every $t \in [0, T_m)$. When $t = 0$ we get $u_0(x) \in W$, which contradicts our assumption $u_0 \in V$. If $I(t_0) = 0$ and $\|u(t_0, \cdot)\|_1\ne 0$, then $u(t_0, x)$ coincides with some ground state solution $\hat{u}(x)$ of (1). Since $\|u(t_0, \cdot)\|_1\ne 0$ from the uniqueness result it follows that $u(t, x) = \hat{u}(x)$ for every $t\geq 0$. Hence $I(u(t, \cdot)) = I(\hat{u}) = 0$ for every $t\geq 0$, which contradicts our assumption $u_0 \in V$. Thus (ii) in Theorem 2.4 is proved.

Remark 1. We rewrite the conservation law (5), (6) by means of (7) in the following way

where from (2) and (3)

Remark 2. If $E(0) = d$ then condition $I(0)<0$ is equivalent to

while condition $I(0)\geq 0$ is equivalent to

For the proofs of our main results in Section 4 and Section 5 we need the following auxiliary statement.

Lemma 2.5. Suppose $u(t, x)$ is the weak solution of (1) - (3) in the maximal existence time interval $[0, T_m)$, $0<T_m \leq \infty$ and $E(0) = d$. If $I(0) <0$, then

The proof of Lemma 2.5 is identical with the proof of Lemma 2.3 in [25] and we omit it.

3.   Improved concavity method of Levine

In the last decades the concavity method, introduced by Levine [15], is one of the powerful methods in the investigation of the finite time blow up of the solutions to nonlinear dispersive equations. The main idea of the concavity method is one to prove finite time blow up of the solutions to the ordinary differential inequality

where $\psi(t)$ is a nonnegative, twice differentiable function for $t>0$. When

then the solution $\psi(t)$ of (14) blows up for a finite time $T^*$ and

In the applications to nonlinear dispersive equations usually $\psi(t)$ is some functional of the solution. For example, $\psi(t) = \int_{R^n} u^2(t, x)dx$ for Klein-Gordon equation. For fourth and sixth order double dispersive equations $\psi(t)$ is more complicated functional, including the $L^2$ norm of the solution and some additional terms.

Let us mention, that condition (15) is only sufficient one for finite time blow up of the solution $\psi(t)$ to (14). The question, which naturally arises, is whether a necessary and sufficient condition for blow up of $\psi(t)$ exists.

In order to give a satisfactory answer of this question, instead of inequality (14) we consider the following nonlinear ordinary differential equation

Here the nonnegative, twice differentiable function $\psi(t)$ is defined in the maximal existence time interval $[0, T_m)$, $0 < T_m \leq \infty$. In the applications to nonlinear dispersive problems, equation (16) naturally appears instead of inequality (14). Since the nonnegative term $Q(t)$ can not be expressed by means of $\psi(t)$, this term has been neglected and (16) has been reduced to (14).

We recall the definition of blow up of a nonnegative function $\psi(t) \in C^2([0, T_m))$ at $T_m$.

Definition 3.1. The nonnegative function $\psi(t) \in C^2([0, T_m))$ blows up at $T_m$ if

Theorem 3.2. Suppose $\psi(t) \in C^2([0, T_m))$ is a nonnegative solution to (16), (17) in the maximal existence time interval $[0, T_m)$, $0< T_m \le \infty$. If $\psi(t)$ blows up at $T_m$, then $T_m < \infty$.

Proof. Step 1. First we will show that

If not, then $\psi'(t) \leq 0$ for every $t\in [0, T_m)$ and the estimate

holds for every $t \in [0, T_m)$. Hence we get

which contradicts (18). Thus (19) holds.

Step 2. Now we will prove that $\psi(t)>0$ for every $t\in [b, T_m)$. From (19) we have that $\psi(b) >0$. Otherwise from (16) it follows that $\gamma \psi'^2(b) \leq 0$, which contradicts (19). In order to prove that $\psi(t)>0$ for every $t \in [b, T_m)$ we suppose by contradiction that there exists $t_0 \in (b, T_m)$ such that

From (16), (17) and (20) we get

i.e. $\psi(t)$ is a convex function for $t\in [b, t_0)$. Hence $\psi'(t) \geq \psi'(b) >0$ for $t\in [b, t_0)$ and $\psi(t)$ is a strictly increasing function for $t\in [b, t_0)$. From the monotonicity of $\psi(t)$ we obtain the following impossible chain of inequalities $0 = \psi(t_0)>\psi(b)>0$. Thus $\psi(t)$ is a positive function satisfying the estimate

Additionally, from (16), (17) and (21) it follows that $\psi(t)$ is a convex function satisfying the inequality

Step 3. Let us prove that $T_m < \infty$. For this purpose we introduce the new function

Straightforward computations give us

and $z(t)$ satisfies the problem

Suppose that $T_m$ is not finite, i.e. $T_m = \infty$. Then from (17), (24) it follows that

Integrating (25) twice from $b$ to $t$ we get

Consequently, there exists a constant $T^*$,

such that $z(T^*) = 0$, or equivalently $\psi(T^*) = \infty$, which contradicts our assumption that $T_m = \infty$. Theorem 3.2 is proved.

The following necessary and sufficient condition for finite time blow up of the solution to the ordinary differential equation (16) is a key result in the investigation of the behavior of the solutions to nonlinear dispersive equations.

Theorem 3.3. Suppose $\psi(t) \in C^2([0, T_m))$ is a nonnegative solution to (16) in the maximal existence time interval $[0, T_m)$, $0< T_m \leq \infty$ and $Q(t) \in C([0, \infty))$, $Q(t) \geq 0$ for $t\geq 0$. Then $\psi(t)$ blows up at $T_m$ if and only if (19) holds, i.e. there exists $b \in [0, T_m)$, such that $\psi'(b)>0$. Moreover, the estimate

holds.

Proof. (Necessity) Suppose $\psi(t)$ blows up at $T_m$. Then (19) holds from Step 1 in the proof of Theorem 3.2, while (27) is a consequence of the inequality (26) in Step 3.

(Sufficiency) Suppose (19) is satisfied. From Step 2 and Step 3 in the proof of Theorem 3.2 it follows that $T_m < \infty$.

If we assume by contradiction that $\psi(t)$ does not blow up at $T_m$, i.e. (18) fails, then

From (22), (28) it follows that $\psi(t)$ is a strictly increasing and bounded function for $t \in [b, T_m)$ so that the limit of $\psi(t)$ for $t \rightarrow T_m$ exists and

Integrating (24) from $b$ to $t <T_m$ we get

or equivalently, from (23)

Thus from (21), (29) and the monotonicity of $\psi'(t)$ we have

The initial value problem

has a classical solution $\varphi(t) \in C^2([T_m, T_m +\delta))$ for sufficiently small $\delta >0$. Hence the function

$\tilde{\varphi}(t) \in C^2([0, T_m +\delta))$, $\tilde{\varphi}(t) \geq 0$ for $t \in [0, T_m +\delta )$ is a classical, nonnegative solution of (16) in the interval $[0, T_m +\delta )$. This contradicts the choice of $T_m$. Hence $\psi(t)$ blows up at $T_m$ and Theorem 3.3 is proved.

4.   Finite time blow up of the solutions to nonlinear Klein-Gordon equation

As a consequence of Theorem 3.2, Theorem 3.3 and Theorem 2.4 we have the following precise results for finite time blow up of the solutions to (1) - (3) in the critical case $E(0) = d$.

Theorem 4.1. Suppose $u(t, x)$ is the weak solution of (1) – (3) with initial energy $E(0) = d$, defined in the maximal existence time interval $[0, T_m)$, $0<T_m \leq \infty$. If $I(0)<0$ and $p_l<(n+4)/n$ then $u(t, x)$ blows up at $T_m$ if and only if

Moreover, $T_m$ is finite, i.e. $T_m<\infty$.

Proof. For the function $\psi(t) = \| u \|^2$, simple computations give us from (10) the identities

Hence $\psi(t)$ satisfies the following ordinary differential equation

where

From (11), (13) in Lemma 2.5 and the Cauchy-Schwartz inequality we have

Thus $\psi(t)$ is a solution to (16), (17) for $\gamma = (p_1+3)/4$ and $Q(t) \geq 0$ defined in (33).

(Necessity) Suppose $u(t, x)$ blows up at $T_m$. From Lemma 2.3 it follows that $\psi(t) = \|u\|^2$ blows up at $T_m$. Then from Step 1 in the proof of Theorem 3.2 for $\psi(t) = \| u \|^2$, $\gamma = (p_1+3)/4$ and $Q(t) \geq 0$ defined in (33) there exists $b \in [0, T_m)$ such that $\psi'(b) = 2 (u(b, \cdot), u_t(b, \cdot))>0$, i.e. (30) is satisfied.

(Sufficiency) Suppose (30) holds, but $u(t, x)$ does not blow up at $T_m$. From Theorem 2.1(ⅰ) it follows that $T_m = \infty$. From (31) and Theorem 2.4(ⅱ) the function $\psi(t) = \|u\|^2$ is a strictly convex one, because $\psi''(t) = 2 \| u_t\|^2 -2 I(t) >0$. Thus (30) gives us the inequality

From (35) there exists $b_1 \in (b, T_m)$, such that $\psi'(b_1)>0$. According to Theorem 3.3 in the interval $[b_1, T_m)$ for $\psi(t) = \|u\|^2$, $\gamma = (p_1+3)/4$ and $Q(t) \geq 0$ defined in (33) it follows that $\|u\|^2$ blows up at $T_m$. Applying Theorem 3.2 we get $T_m<\infty$. Theorem 4.1 is proved.

Remark 3. From the proof of Theorem 4.1 it is clear that the restriction $p_l<(n+4)/n$ for the nonlinear term (2), (3) is used only in the proof of the (Necessity) of Theorem 4.1. Let us note that the statement in the (Sufficiency) of Theorem 4.1 holds for every $p_l$ satisfying (3), i.e. the assumption $p_l<(n+4)/n$ is superfluous.

Remark 4. Let us compare the condition (4) and the new one (30). The careful analysis of the necessary and sufficient conditions (30) in Theorem 4.1 and (4) shows that if (4) holds then (30) is also satisfied at the same time $t = b$. This conclusion follows from (12) in Remark 2 in case $t = b$. Conversely, if (30) holds at $t = b$ then necessarily (4) is satisfied at some time $b_1\geq b$.

In the following theorem we give sufficient conditions for finite time blow up of the solutions to (1) - (3) in terms of the initial data $u_0$, $u_1$.

Theorem 4.2. Suppose $u(t, x)$ is the weak solution of (1) - (3) with initial energy $E(0) = d$, defined in the maximal existence time interval $[0, T_m)$, $0<T_m \leq \infty$. Then the weak solution $u(t, x)$ blows up at $T_m$ when one of the following conditions is fulfilled:

(ⅰ) $(u_0, u_1)\geq 0$ and $I(0)<0$;

(ⅱ)

Moreover, $T_m$ is finite, i.e. $T_m <\infty$.

Proof. (ⅰ) The proof of Theorem 4.2 (ⅰ) follows immediately from the sufficiency part of Theorem 4.1 and Remark 3 for $b = 0$ when $(u_0, u_1)>0$. If $(u_0, u_1) = 0$ then from (31) and Theorem 2.4 we obtain that $\psi''(0) = 2 \| u_1\|^2 -2 I(0)>0$ and $\psi'(b)>\psi'(0)\geq 0$ for every $b>0$. Since $\psi(t)$ satisfies (32) and (34) from Theorem 3.3 it follows that $\psi(t)$ blows up for a finite time, i.e. $u(t, x)$ blows up for a finite time.

(ⅱ) Suppose (36) hold. Since

from Remark 2 it follows that $I(0)<0$.

In order to prove statement (ⅱ) we suppose by contradiction that $u(t, x)$ does not blow up at $T_m$. Then from the local existence result, Theorem 2.1, it follows that $T_m = \infty$. Thus $u(t, x)$ is globally defined for every $t\geq 0$.

If (30) is satisfied, i.e. there exists $b\in [0, T_m)$ such that $( u(b, \cdot), u_t (b, \cdot))\geq 0$, then from Theorem 4.1 $u(t, x)$ blows up at $T_m$, which contradicts our assumption. Hence $u(t, x)$ blows up at $T_m$ and from Theorem 3.2 it follows that $T_m<\infty$. Thus statement (ⅱ) is proved when (30) is fulfilled.

If (30) does not hold, then

From (31) the function $\psi(t) = \| u \|^2$ is a solution to the equation

Here $\alpha = p_1-1>0$, $\beta = 2(p_1+1)E(0)>0$ and

because $B(t)$, given in (11), is a non negative function. Equation (38) has a unique classical solution

and

From (39) and (40) we get

By means of (31) the function $h(t) = -(u, u_t)$ satisfies the equation

Since

we have the estimates

After the limit $t\to \infty$ in the above inequality, from (37), we get the inequalities

Thus we obtain

because $\varepsilon$ is an arbitrary positive constant.

Since $\psi(t)$ is monotone decreasing and bounded from below with zero, after the limit $t\to\infty$ in (41) we get from (42) the identity

Hence necessarily we have

and from L'Hospital's rule it follows that

i.e.

Multiplying (38) with $\psi'(t)$ and integrating from $0$ to $t$ we obtain the identity

Since $\beta = 2(p_1+1)d$, $\alpha = p_1-1$, $\psi(0) = \|u_0\|^2$ and $\psi'(0) = 2(u_0, u_1)$, then the second inequality in (36) can be rewritten as

Thus from (37) it follows that $\psi'(0)<0$ and (46) gives us

Let us consider the case

Since $\psi(t)$ is a strictly decreasing function for $t\in[0, \infty)$, from (47) and (48), there exists a point $t_1$, $t_1\in(0, \infty)$, such that

Then for $t = t_1$ in (45) we get

Now we consider the case

After the limit $t\to\infty$ in (45) the equality (45) becomes

In both cases from (49) and (50) we have

The above inequality is satisfies if

or equivalently

which contradicts condition (36). Thus $u(t, x)$ blows up at $T_m$ and from Theorem 3.2 it follows that $T_m<\infty$. Theorem 4.2 is proved.

Remark 5. The statement of Theorem 4.2(ⅰ) has been already proved in a different way for the nonlinear wave equation in a bounded domain, see e.g. [9,26] and for nonlinear Klein-Gordon equation, see [16,19]. In the present paper the proof of Theorem 4.2(ⅰ) is a consequence of Theorem 3.3.

Remark 6. Let the initial data satisfy conditions (36). Then from (12) it follows, that $I(0)<0$, i.e the assumption $I(0)<0$ is unnecessary in Theorem (4.2)(ⅱ).

In the following corollary we reformulate the statements in Theorem 4.2. The requirement for the sign of the Nehari functional $I(0)$ is replaced by the assumptions on the initial data according to Remark 2.

Corollary 1. Suppose $u(t, x)$ is the weak solution of (1) - (3) with initial energy $E(0) = d$, defined in the maximal existence time interval $[0, T_m)$, $0<T_m \leq \infty$. Then the weak solution $u(t, x)$ blows up at $T_m$ when the initial data satisfy one of the following conditions:

(ⅰ)

(ⅱ)

Moreover, $T_m$ is finite, i.e. $T_m <\infty$.

Below we compare the result in Theorem 4.2 (Corollary 1) with the result in [16] for the nonlinear term

Proposition 1. Suppose $u(t, x)$ is the weak solution of (1) with initial energy $E(0) = d$, defined in the maximal existence time interval $[0, T_m)$, $0<T_m \leq \infty$ and $f(u) = u^2+u^3$. Then $u(t, x)$ blows up at $T_m<\infty$ when one of the following conditions holds:

(ⅰ) (Theorem (4.2)(i), [16,Theorem 1.3(3)])

(ⅱ) (Theorem (4.2)(ii))

(ⅲ) ([16,Theorem 1.3(3)])

where $u_e$ satisfies conditions $I(u_e) = 0$ and $J(u_e) = d$.

Proof. (ⅰ) and (ⅱ) We apply Theorem 4.2 for $p_1 = 2$, $p_2 = 3$, $a_1 = 1$, $a_2 = 1$, $a_k = 0$ for $k = 3, \ldots, l$, $b_j = 0$ for $j = 1, \ldots, s$, $n\leq 3$ and

According to Theorem 4.2 and Corollary 1 the solution $u(t, x)$ of (1), (51) blows up for a finite time when the initial data satisfy one of the following conditions:

So the statements (ⅰ) and (ⅱ) are proved. Note, that for $(u_0, u_1)\geq 0$ and $I(0)<0$ the result in Theorem 1.3(3) in [16] coincides with the statement in Proposition 1(i).

(ⅲ) For $(u_0, u_1)<0$ Theorem 1.3(3) says that the solution blows up for finite time if

where $u_e$ satisfies conditions $I(u_e) = 0$ and $J(u_e) = d$. Since the conservation law (6) gives us

assumptions (55) are equivalent to

The above inequality holds only for

When the opposite inequality is satisfied, i.e.

then the set of functions satisfying Theorem 1.3(3) in [16] is empty. In this case the finite time blow up of the solutions is possible only under conditions of Theorem (4.2)(ⅱ), i.e. when (54) is satisfied.

However, if (52) holds, then the conditions for finite time of the solution to (1), (51) in Theorem (4.2)(ⅱ) and Theorem 1.3(3) in [16] are completely different. Indeed from the inequality

it follows that the intervals for $\|u_0\|^2$ in assumptions (54) and (53) have no intersection points. Thus the result in Theorem (4.2)(ⅱ) is a new one.

5.   Global existence and the asymptotic behavior of the solutions to the nonlinear Klein-Gordon equation

5.1.   Global existence of the solutions to the nonlinear Klein-Gordon equation

Theorem 5.1. Suppose $u(t, x)$ is the weak solution of (1) - (3) with initial energy $E(0) = d$, defined in the maximal existence time interval $[0, T_m)$, $0<T_m \leq \infty$, $p_l<(n+4)/n$ and $I(0)< 0$. Then $u(t, x)$ is globally defined for every $t \geq 0$, i.e. $T_m = \infty$, if and only if

Proof. (Necessity) Suppose $u(t, x)$ is defined for every $t\geq 0$, i.e. $T_m = \infty$. If (56) fails, then there exists $b\in[0, \infty)$ such that

From Theorem 4.1 it follows that $u(t, x)$ blows up for finite time $T_m<\infty$, which contradicts our assumption $T_m = \infty$.

(Sufficiency) Suppose condition (56) is satisfied. Then $\|u(t, \cdot)\|$ is a strictly decreasing function and the following inequality holds

From Lemma 2.3 it follows that

and from the local existence result we get $T_m = \infty$. Thus $u(t, x)$ is defined for every $t\geq 0$ and Theorem 5.1 is proved.

Remark 7. The growth condition $p_l<(n+4)/n$ for the nonlinear term (2), (3) is used only in the proof of the (Sufficiency) of Theorem 5.1. For the proof of the (Necessity) of Theorem 5.1, assumption $p_l<(n+2)/(n-2)$ in (3) is enough.

Let us formulate necessary conditions on the initial data for global existence of the solutions to (1) - (3).

Theorem 5.2. Suppose $u(t, x)$ is the weak solution of (1) - (3) with initial energy $E(0) = d$, defined in the maximal existence time interval $[0, T_m)$, $0<T_m \leq \infty$.

(ⅰ) If $I(0)\geq 0$ then $u(t, x)$ is globally defined for every $t \geq 0$, i.e. $T_m = \infty$;

(ⅱ) If $I(0)<0$, then a necessary condition for global existence of $u(t, x)$ for every $t\geq 0$ is

Proof. The statement (ⅰ) in Theorem 5.2 has been already proved in [9,16,24,27] when $u_0 \in W = \left\{ u \in H^1( \mathbb{R}^n); I(0)>0 \right\} \cup\left\{ 0\right\}$. Since the condition $I(u_0) \ge 0$ is slightly more general then $u_0 \in W$, for completeness we give the proof.

(ⅰ) If $I(0) = 0$ and $\|u_0\|_1\ne 0$ then $u_0\in \mathcal{N}$ and $J(u_0)\geq d$. From (6) and (7) we get

i.e. $\|u_1\| = 0$ and $J(u_0) = d$. Since the function $u_0(x)$ is a solution to (1) - (3), from the uniqueness result it follows that $u(t, x) = u_0(x)$ for every $t\geq 0$, i.e. $u(t, x)$ is a global solution.

If $I(0)>0$ or $I(0) = 0$ but $\|u_0\|_1 = 0$, i.e. $u_0 = 0$ for a.e. $x\in \mathbb{R}^n$, then $u_0 \in W$ and from Theorem 2.4 $I(t) \in W$ for every $t \in [0, T_m)$. From the conservation law (5), see also (10), we have the estimate

Thus from statement (ⅰ) in the local existence result Theorem 2.1 it follows that $T_m = \infty$.

(ⅱ) Suppose that $u(t, x)$ is defined for every $t \ge 0$, i.e. $T_m = \infty$. If (57) fails, i.e. one of the following conditions is satisfied,

or

then from Theorem 4.2 it follows that $u(t, x)$ blows up for finite time. Thus $T_m <\infty$, which contradicts our assumption $T_m = \infty$. Theorem 5.2 is proved.

5.2.   Asymptotic behavior of the global solutions to the nonlinear Klein-Gordon equation

Let us mention that the asymptotic behavior of the global solution to the nonlinear wave equation in bounded domains has been studied in [9] and to the Klein-Gordon equation with quadratic-cubic nonlinearity - in [16].

In the following theorem we study the asymptotic behavior for $t\to\infty$ of the global solutions to (1) - (3) with critical initial energy $E(0) = d$ and $I(0)<0$, i.e. when condition (ⅱ) of Theorem 5.2 is satisfied.

Theorem 5.3. Suppose $u(t, x)$ is a weak solution of (1) - (3) with initial energy $E(0) = d$, defined for every $t\in [0, \infty)$, $p_l<(n+4)/n$ and $I(0)<0$. Then there exist a sequence of time $t_m\to\infty$, functions $\hat{u}(x)\in \mathrm{H}^1(\mathbb{R}^n)$ and $\hat{v}(x)\in \mathrm{L}^2(\mathbb{R}^n)$, such that $u(t_m, x)\rightharpoonup \hat{u}(x)$ weakly in $\mathrm{H}^1(\mathbb{R}^n)$ and $u_t(t_m, x)\rightharpoonup \hat{v}(x)$ weakly in $\mathrm{L}^2(\mathbb{R}^n)$ when $m\to\infty$. Moreover,

(ⅰ) $\|\hat{v} \| = 0$ and $\liminf\limits_{t\to\infty} \|u_t(t, \cdot)\| = 0;$

(ⅱ) $\lim\limits_{t\to\infty} I(u(t, \cdot)) = 0;$

(ⅲ) $\lim\limits_{t\to\infty} t (u(t, \cdot), u_t(t, \cdot)) = 0.$

Proof. Since the solution $u(t, x)$ is defined for every $t$, it follows that

Indeed, if (58) fails, then from the local existence result, Theorem 2.1 (ⅰ), we get $\lim_{t\to\infty}\|u(t, \cdot)\|_1 = \infty$, which contradicts the results in Lemma 2.3 and Theorem 3.2. From the embedding of $\mathrm{H}^1(\mathbb{R}^n)$ into $\mathrm{L}^p(\mathbb{R}^n)$ for $p>2$ we obtain

and from the conservation law (6)

where the constants $C_1$ and $C_2$ depend on the parameters of the nonlinearity $f(u)$ as well as on the initial data. As a consequence of (58), (59) and (60), there exist a sequence $t_m\to \infty$ and functions $\hat{u}(x)\in \mathrm{H}^1(\mathbb{R}^n)$ and $\hat{v}(x)\in \mathrm{L}^2(\mathbb{R}^n)$, such that for every $\Phi\in \mathrm{H}^1(\mathbb{R}^n)$ and every $\Theta\in \mathrm{L}^2(\mathbb{R}^n)$ the following equations are true

i.e. $u(t, x)\rightharpoonup \hat{u}(x)$ and $u_t(t, x)\rightharpoonup \hat{v}(x)$ for $t \rightarrow \infty$.

(ⅰ) We will prove that $\| \hat{v}\| = 0$. Since $I(t)<0$ for every $t\ge 0$ (see Theorem 2.4), integrating (31) for $s>t\geq0$ we obtain

After the change of the variable $\tau = \lambda+t$ (63) becomes

Thus for every $s_m\to \infty$ from (42) and (64) it follows that

As a consequence of Fatou's lemma we get

By means of the weak convergence of $u_t(t, x)$ to $\hat{v}(x)$ in $\mathrm{L}^2(\mathbb{R}^n)$, i.e. (61), and the lower semicontinuity of the norm of $\mathrm{L}^2(\mathbb{R}^n)$, we get for some $\lambda_0\in[0, 1]$, $s_m = t_m-\lambda_0$ and (65) the final inequality

i.e.

Thus we proved that every weak limit of $u_t(t_m, x)$ for $m\to \infty$ is zero and

(ⅱ) From (44) and (38) it follows that

Thus (31) and (ⅰ) in Theorem 5.3 give us

Hence $\lim_{t\to\infty} I(t) = 0$ and (ⅱ) is proved.

(ⅲ) From (42) and (62) after the limit $s \rightarrow \infty$ we obtain

and integrating (66) from $0$ to $\infty$ it follows that

Applying Fubini's theorem, (67) becomes

After the integration of (66) from $0$ to $s$ we have

and Fubini's theorem implies

After the limit $s \rightarrow \infty$ in (69), from (66) and (68) it follows that

hence (ⅲ) holds. Theorem 5.3 is proved.

Acknowledgments

The all authors have been partially supported by the National Scientific Program "Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICTinSES)", contract No D01205 / 23.11.2018, financed by the Ministry of Education and Science in Bulgaria. The first author has been also supported by the Bulgarian National Science Fund under grant DFNI 12/5. The second author has been also supported by the Bulgarian National Science Fund under grant K$\Pi$-06-H22/2.