A Sufficient Condition for Blowup of the Nonlinear Klein-Gordon Equation with Positive Initial Energy in FLRW Spacetimes

The nonlinear Klein-Gordon equations are a class of important evolution equations that describe the movement of spinless relativistic particles, which can lend understanding in many physical applications (see [8] and references therein). In this paper we consider the question of finite time blow up for a nonlinear class of Klein-Gordon equations propagating on a fixed cosmological spacetime. We have adapted the methods of Yang & Xu in [8] and Wang in [7], both of which prove blowup results for solutions to the Klein-Gordon equation in a non-expanding spacetime. The result from [7] was the first to show a condition for blowup using this general method, and this work was expanded upon in [8] to broaden the conditions that reliably lead to blowup. Both of these were done under the assumption of a Minkowski background metric – for which energy conservation for Klein-Gordon is readily available. On the other hand, our work focuses on adapting this method of proof to backgrounds that include expanding spacetimes – for which only an energy inequality is available. Due to the accelerating nature of the expansion of the observable universe, such cosmological spacetimes are of general interest. The results in this paper also supplement those in [2]. In that work, the authors establish small data global existence for a class of semilinear Klein-Gordon equations propagating in a family of expanding cosmological backgrounds which include the de Sitter spacetime. On the other hand, in this paper we are considering large initial data satisfying a specific set of conditions that ensure $L^{2}$ blow up in finite time.

The general concavity method employed to prove the blowup result in this paper, along with in [7] and [8], is based on the work of Levine in [3] and [4], which has been a common wellspring for such results. This method relies on identifying time invariant spaces, then proving that solutions in this invariant space must blowup based on consistent negative concavity and slope of a functional based on this solution. As mentioned, we employ this method to classes of solutions of the Klein-Gordon equation, though in more generality than [8] and [7] by considering all metrics that have a certain restriction on the expanding factor. The fact that there must be restrictions on the expanding factor should satisfy intuition that aggressively expanding spacetimes would create problems with local existence for fluids and would smooth out initial data too quickly to allow for blowup. Furthermore, this method requires some assumptions on the initial data relative to the initial energy which is necessary for the concavity argument and can be broken down into cases (listed in Table 1 below). This table is adapted from Yang & Xu in [8] to illustrate our results as well as the remaining unsolved problems in the context of cosmological spacetimes. Note that our assumption (2.13) follows directly from Cases II & III.


Table 1: Obtained results and unsolved problems.

Case I	Initial data leading to high energy blowup of problem (2.4)
	$I\left(u_{0}\right)<0$	$\left\|u_{0}\right\|^{2}\geq\frac{3(\epsilon+2)}{\tilde{m}^{2}\epsilon}E(t_{0})$	$\frac{3(\epsilon+2)}{\tilde{m}^{2}\epsilon}E(t_{0})>\left(u_{0},u_{1})\right.>0$	Still open
Case II	$I\left(u_{0}\right)<0$	$\left\|u_{0}\right\|^{2}\geq\frac{3(\epsilon+2)}{\tilde{m}^{2}\epsilon}E(t_{0})$	$\left(u_{0},u_{1}\right)\geq\frac{3(\epsilon+2)}{\tilde{m}^{2}\epsilon}E(t_{0})>0$	Solved in this paper
Case III	$I\left(u_{0}\right)<0$	$\frac{3(\epsilon+2)}{\tilde{m}^{2}\epsilon}E(t_{0})>\left\|u_{0}\right\|^{2}$	$\left(u_{0},u_{1}\right)\geq\frac{3(\epsilon+2)}{\tilde{m}^{2}\epsilon}E(t_{0})>0$	Solved in this paper
Case IV	$I\left(u_{0}\right)<0$	$\frac{3(\epsilon+2)}{\tilde{m}^{2}\epsilon}E(t_{0})>\left\|u_{0}\right\|^{2}$	$\frac{3(\epsilon+2)}{\tilde{m}^{2}\epsilon}E(t_{0})>\left(u_{0},u_{1}\right.)>0$	Still open

As our work follows directly from the method in [8], we do not have any specific conditions on the $||u_{0}||^{2}$ term. Their process removed specific consideration of this term that had been present in [7], and so covered the Case II and Case III with this value having no specific relationship to the bounding constant. So while a lower bound on this quantity was required for [7], our method has no such requirement and therefore covers the two cases.

Let $n\in\mathbb{N}$ and $(\mathcal{M},{\mathbf{g}})$ be a Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime with topology $\mathbb{R}^{+}\times\mathbb{R}^{n}$ and metric

(2.1)

$${\mathbf{g}}:=-dt^{2}+a(t)^{2}\big{(}dx_{1}^{2}+dx_{2}^{2}+...+dx_{n}^{2}\big{)}.$$

Let $m\neq 0$ and $\Box_{g}$ be the covariant wave operator associated to $\mathbf{g}$ defined by

(2.2)

$$\square_{{\mathbf{g}}}u:=\frac{1}{\sqrt{|{\mathbf{g}}|}}\partial_{\alpha}\left({\mathbf{g}}^{\alpha\beta}\sqrt{|{\mathbf{g}}|}\partial_{\beta}u\right),$$

where $|{\mathbf{g}}|=-\det({\mathbf{g}}_{\alpha\beta})$, ${\mathbf{g}}^{\alpha\beta}$ are the components of the inverse of ${\mathbf{g}}_{\alpha\beta}$, and greek indices run from $0$ to $n$. For the FLRW metric (2.1),we have

(2.3)

$$\square_{{\mathbf{g}}}u=-u_{tt}-n\frac{{\dot{a}}}{a}u_{t}+\frac{1}{a^{2}}\Delta u,$$

with $\Delta$ denoting the Laplacian operator on the $n$-dimensional flat metric

$$\Delta:=\sum_{i=1}^{n}\frac{\partial^{2}}{\partial x^{2}_{i}}.$$

Let $0<t_{0}<T$, $t\in[t_{0},T)$, and $x\in\mathbb{R}^{n}$. In this work we study solutions to the Cauchy problem for the nonlinear Klein-Gordon equation

(2.4)

$$\begin{cases}&-\square_{{\mathbf{g}}}u+m^{2}u=f(u)\\ &u(t_{0},x)=u_{0}(x),\ u_{t}(t_{0},x)=u_{1}(x).\end{cases}$$

where $u_{0}\in H_{0}^{1}(\mathbb{R}^{n})$ and $u_{1}\in L^{2}(\mathbb{R}^{n})$. This Cauchy problem describes a local self-interaction for a massive scalar field. In quantum field theory the matter fields are described by a function $u:[t_{0},T)\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ that must satisfy equations of motion. In the case of a massive scalar field, the equation of motion is the semilinear Klein-Gordon equation in (2.4).

We now specify the assumptions in our work.

In this section we make use of the local wellposedness result in [2, Section 1]. We state the theorem, hypotheses, and (readjusted) notation for the reader’s convenience here.

The statement of this theorem, combined with our assumptions in Section 2 and with the fact that $H_{0}^{1}(\mathbb{R}^{n})\subset H^{1}(\mathbb{R}^{n})$ immediately implies that the Cauchy problem (2.4) satisfies the conditions of this theorem with $\Gamma(t)=1$.

In this section we develop the framework needed to prove our energy estimate. We largely follow the setup of [5] and [6]. Define the energy-momentum tensor to be

(4.1)

$$T_{\alpha\beta}:=\partial_{\alpha}u\partial_{\beta}u-\frac{1}{2}g_{\alpha\beta}\Big{(}\partial^{\mu}u\partial_{\mu}u+m^{2}u^{2}-2F(u)\Big{)}\;.$$

Let $D$ be the covariant derivative corresponding to the metric $g$. The energy momentum tensor is divergence free

$$D^{\alpha}T_{\alpha\beta}=0.$$

Given a (smooth) vector field $X$ we define the 1-form

$${}^{(X)}P_{\alpha}:=T_{\alpha\beta}X^{\beta}.$$

Taking the divergence yields

(4.2)

$$D^{\alpha}\big{(}{}^{(X)}P_{\alpha}\big{)}=\frac{1}{2}\;^{(X)}\pi^{\alpha\beta}T_{\alpha\beta}$$

where the symmetric 2-tensor ${}^{(X)}\pi$ is the deformation tensor of $g$ with respect to $X$. Integrating this divergence over the spacetime region

$$\{(t,x)\;|\;t_{0}\leq t\leq t_{1}\}$$

and using Stokes’ theorem we get the following multiplier identity

(4.3)

$$\begin{split}\int_{t=t_{0}}\;^{(X)}P_{\alpha}N^{\alpha}|{\mathbf{g}}|^{\frac{1}{2}}\ dx-\int_{t=t_{1}}\;^{(X)}P_{\alpha}N^{\alpha}|{\mathbf{g}}|^{\frac{1}{2}}\ dx=\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{n}}(\frac{1}{2}\;^{(X)}\pi^{\alpha\beta}T_{\alpha\beta})\ dV_{g}\end{split}$$

where

$$dV_{g}:=|{\mathbf{g}}|^{\frac{1}{2}}dtdx$$

and $N$ is the vector field defined below:

(4.4)

$$N^{\alpha}:=\frac{-g^{\alpha\beta}\partial_{\beta}t}{(-g^{\alpha\beta}\partial_{\alpha}t\partial_{\beta}t)^{\frac{1}{2}}}=-g^{\alpha 0}.$$

The integrand on the left hand side of Eq. (4.3) is the energy density associated to $X$ through the foliation by the spacelike hypersurfaces $t=const$.

Given a vector field $X$ we define the normalized deformation tensor of $X$ to be

(4.5)

$${}^{(X)}\hat{\pi}=\;^{(X)}\pi-\frac{1}{2}|{\mathbf{g}}|\cdot trace(^{(X)}\pi)$$

which can be computed using the following identity (see [5]):

(4.6)

$$^{(X)}\hat{\pi}^{\alpha\beta}=-|{\mathbf{g}}|^{-\frac{1}{2}}X(|{\mathbf{g}}|^{\frac{1}{2}}g^{\alpha\beta})-g^{\alpha\beta}\partial_{\gamma}X^{\gamma}+g^{\alpha\gamma}\partial_{\gamma}X^{\beta}+g^{\beta\gamma}\partial_{\gamma}X^{\alpha}.$$

In combination with Eq. (4.2), we get

(4.7)

$$D^{\alpha}\big{(}{}^{(X)}P_{\alpha}\big{)}=\frac{1}{2}\;^{(X)}\hat{\pi}^{\alpha\beta}\partial_{\alpha}u\partial_{\beta}u.$$

Combining (4.7) and (4.3) yields

(4.8)

$$\int_{t=t_{0}}\;^{(X)}P_{\alpha}N^{\alpha}|{\mathbf{g}}|^{\frac{1}{2}}dx-\int_{t=t_{1}}\;^{(X)}P_{\alpha}N^{\alpha}|{\mathbf{g}}|^{\frac{1}{2}}dx=\int_{t_{0}}^{t_{1}}\int_{\mathbb{R}^{n}}\frac{1}{2}\Big{(}\;^{(X)}\hat{\pi}^{\alpha\beta}\partial_{\alpha}u\partial_{\beta}u\Big{)}|{\mathbf{g}}|^{\frac{1}{2}}dtdx\\$$

In this section we will prove the energy nonincreasing property as stated below:

Hence,

(5.1)

$$\frac{1}{2}\big{(}\;^{(X)}\hat{\pi}^{\alpha\beta}\partial_{\alpha}u\partial_{\beta}u\big{)}\ |{\mathbf{g}}|^{\frac{1}{2}}=n\frac{\dot{a}}{a}|u_{t}|^{2}+n\frac{\dot{a}}{a^{3}}|\nabla u|^{2}\geq 0.$$

From Eq. (4.8) it follows that:

(5.2)

$$\begin{split}\int_{t=t_{0}}\;^{(X)}P_{\alpha}N^{\alpha}|{\mathbf{g}}|^{\frac{1}{2}}dx&-\int_{t=t_{1}}\;^{(X)}P_{\alpha}N^{\alpha}|{\mathbf{g}}|^{\frac{1}{2}}dx\geq 0.\end{split}$$

Expanding the integrand yields

$${}^{(X)}P_{\alpha}N^{\alpha}|{\mathbf{g}}|^{\frac{1}{2}}=-g^{\alpha 0}T_{\alpha 0}a^{-n}a^{n}=T_{00}=\frac{1}{2}\Big{(}|u_{t}|^{2}+a^{-2}|\nabla u|^{2}+m^{2}u^{2}-2F(u)\Big{)}$$

Combining this with Eq. (5.2) gives us

(5.3)

$$E(t_{0})\geq E(t)\;\;\;\;\forall t\geq t_{0}.\;\;\;\;\;\;\;\square$$

In this section we will prove that the set $\mathcal{B}$ defined in Eq. (2.12) is invariant for the Cauchy problem when the assumptions in the main theorem are satisfied.