Finite time blow-up of semi-linear Klein-Gordon equations
with positive initial energy in FLRW spacetimes

We consider the Cauchy problem of semi-linear Klein-Gordon equations in Friedmann-Lemaître-Robertson-Walker spacetimes (FLRW spacetimes for short). FLRW spacetimes are solutions of the Einstein equations with the cosmological constant under the cosmological principle. They describe the spatial expansion or contraction, and yield some important models of the universe. Let $n\geq 1$ be the spatial dimension, $a(\cdot)$ be a scale-function defined on an interval $[0,T_{0})$ for some $0<T_{0}\leq\infty$, $c>0$ be the speed of light. The metrics $\{g_{\alpha\beta}\}$ of FLRW spacetimes are expressed by

where we have put the spatial curvature as zero, the variable $\tau$ denotes the proper time, $x^{0}=t$ is the time-variable (see e.g., [2, 3]). When $a(\cdot)=1$ or $a(t)=e^{Ht}$ for the Hubble constant $H\in{\mathbb{R}}$, the spacetime with (1.1) reduces to the Minkowski spacetime or the de Sitter spacetime, respectively.

We recall the derivation of the Klein-Gordon equation. We denote the first and second derivatives of one variable function $a$ by $\dot{a}$ and $\ddot{a}$. The Klein-Gordon equation generated by the above metric $(g_{\alpha\beta})_{0\leq\alpha,\beta\leq n}$ is given by $-(\sqrt{|g|})^{-1}\partial_{\alpha}(\sqrt{|g|}g^{\alpha\beta}\partial_{\beta}u)+m^{2}u=f(u)$ for the determinant $g:=\det(g_{\alpha\beta})$ and the inverse matrix $(g^{\alpha\beta})$, i.e.,

where $\Delta:=\sum_{j=1}^{n}\partial_{j}^{2}$ denotes the Laplacian.

In this paper, we consider the Cauchy problem of (1.2) given by

for $(t,x)\in[t_{0},T)\times{\mathbb{R}}^{n}$ with $0\leq t_{0}<T\leq T_{0}$, where $u_{0}$, $u_{1}$ are given initial data. For semi-linear terms, we assume that $f:{\mathbb{C}}\rightarrow{\mathbb{C}}$ satisfies that there exists $F:{\mathbb{C}}\rightarrow{\mathbb{R}}$ and $\varepsilon>0$ such that

for any $u:[t_{0},T)\rightarrow{\mathbb{C}}$. We note that if we set $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$, then the condition on (1.4) is described as

for $u:[t_{0},T)\rightarrow{\mathbb{R}}$, which has been considered in [8, 14]. The assumption (1.4) includes the power-type non-linearity. If we take $f(u)=|u|^{p-1}u$ for $u\in{\mathbb{C}}$ with $p>1$, then we have to set $\varepsilon\leq p-1$. If we take $f(u)=-|u|^{p-1}u$ for $u\in{\mathbb{C}}$ with $p>1$, then we have to set $\varepsilon\geq p-1$. For another example, we note that $f(u)=\pm|u|^{p}$ for $u\in{\mathbb{R}}$ with $p>1$, then we have to set $\varepsilon=p-1$.

We denote the Lebesgue space by $L^{q}(I)$ for an interval $I\subset{\mathbb{R}}$ and $1\leq q\leq\infty$ with the norm

We denote by $\|\cdot\|$ the $L^{2}({\mathbb{R}^{n}})$ norm, and by $\|\cdot\|_{q}$ the $L^{q}({\mathbb{R}^{n}})$ norm for $q\neq 2$. We also denote $(u,v):=\int_{\mathbb{R}^{n}}u\overline{v}dx$ by the $L^{2}$-inner product.

We define the energy functional

the Nehari functional

and the unstable set

for the Cauchy problem of (1.3). We note that the Nehari functional $I(u)$ depends on time. The set $\mathcal{B}$ is invariant under our assumptions for initial data. More precisely, if $I(u_{0})<0$ holds, then we have $u\in\mathcal{B}$ (see, Lemma 2.3 and Lemma 2.4, below).

In [9], the existence of global solutions was proved for semi-linear Klein-Gordon equations in de Sitter spacetime with $n\leq 4$. This result was extended to the case of general FLRW spacetimes in [4, 11]. Galstian and Yagdjian in [4] also proved the local well-posedness with $n\geq 1$ for semi-linear Klein-Gordon equations in FLRW spacetimes for any $u_{0}\in H^{1}({\mathbb{R}^{n}})$, $u_{1}\in L^{2}({\mathbb{R}^{n}})$ with $\dot{a}(\cdot)>0$.

The nonexistence of the global solution of the Klein-Gordon equation in the Minkowski spacetime (i.e., $\dot{a}\equiv 0$) was shown by Levine in [6] when the initial energy was negative. We also refer to the result [15], which established a sharp condition for blowup of one in the Minkowski spacetime when the initial energy had an upper bound. When the initial energy was arbitrarily high, the first result was a condition which Wang in [12] showed for blowing-up solutions corresponding to the Klein-Gordon equation in the Minkowski spacetime. This result was extended by Yang and Xu in [14].

On the other hand, for FLRW spacetimes, the case of the gauge invariant semi-linear term of the form $f(u)=\lambda|u|^{p-1}u$ with $1<p<\infty$, $\lambda\in{\mathbb{C}}$ was considered in [11] for $\dot{a}\leq 0$ by the concavity method for $a^{2}\|u\|^{2}$. Moreover, McCollum, Mwamba and Oliver in [8] showed a condition for blowing-up solutions of (1.3) for large initial data with the positive initial energy.

Our work is to solve Case ${\rm{I}}$, Case ${{\rm{I}\rm{I}}}$ and Case ${{\rm{I}\rm{I}\rm{I}}}$ in Table 1 under $\widetilde{m}>0$ and $\widetilde{c}>0$ defined by (1.16), below. Case ${{\rm{I}\rm{V}}}$ is still open.

We show the following theorem and corollary for blowing-up solutions of (1.3) for large initial data. Put $a_{0}:=a(0)$ and $a_{1}:=\dot{a}(0)$.

When $f\in C({\mathbb{C}},{\mathbb{C}})$ satisfies (1.11) with (1.12), we note that the condition (1.14) does not hold for sufficiently small data (see, Appendix in [8]).

In [8], a sufficient condition for blowing-up solutions of (1.3) was shown under $I(u_{0})<-n\dot{a}a^{-1}\widetilde{m}\widetilde{c}\operatorname{Re}(u_{0},u_{1})<0$ and $\frac{\widetilde{m}^{2}\widetilde{c}^{2}\varepsilon}{3(\varepsilon+2)}\operatorname{Re}(u_{0},u_{1})>E(t_{0})\geq 0$ with $\widetilde{m},\,\widetilde{c}>0$. Since we have shown blowing-up solutions of (1.3) under $I(u_{0})<0$ and $\frac{\widetilde{m}^{2}\widetilde{c}^{2}\varepsilon}{2(\varepsilon+2)}\operatorname{Re}(u_{0},u_{1})>E(t_{0})\geq 0$ with $\widetilde{m},\,\widetilde{c}>0$ in Theorem 1.3, we improve the result in [8] in terms of the upper bounds of $I(u_{0})$ and $E(t_{0})$. Moreover, we also have shown the blowing-up solution of (1.3) under $I(u_{0})<0$ and $\frac{\widetilde{m}^{2}\widetilde{c}^{2}\varepsilon}{2(\varepsilon+2)}\|u_{0}\|>E(t_{0})\geq 0$. Blowup of (1.3) in this condition was not shown in [8].

We adapt the concavity method which was first introduced by Levine in [6, 7]. In the Minkowski spacetime, this method was employed for the function $m^{2}\|u\|^{2}$ in [6, 7, 12, 14]. McCollum, Mwamba and Oliver in [8] also employed this method for $m^{2}\|u\|^{2}$ in FLRW spacetimes. Instead of the function $m^{2}\|u\|^{2}$, we employ the concavity method for a new function $\theta$ given by

for $t\in[t_{0},T)$ with some $T\in(t_{0},T_{0}]$, where

By the second and third terms of $\theta$, we improve the condition on $I(u_{0})$, and we solve Case ${\rm{I}}$. Moreover, we can consider the condition on $m=0$ and $E(t_{0})<0$ in Theorem 1.1 and Theorem 1.3. This yields that our results also include blowing-up solutions for semi-linear wave equations in FLRW spacetimes with the negative energy. Setting the third term of $\theta$ is inspired by [5] and [13]. Gazzola and Squassina in [5], and Xu and Ding in [13] showed some conditions for blowing-up solutions of damped wave, and Klein-Gordon equations in the Minkowski spacetime.

Now, we introduce some concrete examples of the scale-function $a$. For $\sigma\in\mathbb{R}$ and the Hubble constant $H\in{\mathbb{R}}$, we put

and define $a(\cdot)$ by

for $0\leq t<T_{0}$. We note $a_{0}=a(0)$ and $H=\dot{a}(0)/a(0)$, where $\dot{a}:=da/dt$. This scale-function $a(\cdot)$ describes the Minkowski spacetime when $H=0$ (namely, $a(\cdot)$ is a constant $a_{0}$), the expanding space when $H>0$ with $\sigma\geq-1$, the blowing-up space when $H>0$ with $\sigma<-1$ (the “Big-Rip” in cosmology), the contracting space when $H<0$ with $\sigma\leq-1$, and the vanishing space when $H<0$ with $\sigma>-1$ (the “Big-Crunch” in cosmology). It describes the de Sitter spacetime when $\sigma=-1$ (see, e.g., [10]).

We obtain the following corollaries from the above theorems, respectively for the concrete example of $a$ given by (1.18).

In Corollaries 1.5 and 1.6, the case $H=0$ reduces to the semi-linear Klein-Gordon equation in the Minkowski spacetime, which was extensively studied (see, e.g., [1, 6, 12, 13, 14, 15] on blowing-up solutions, and the references therein). We included this case to compare it with the case $H\neq 0$.

This paper is organized as follows. In Section 2, we collect fundamental properties on some ordinary differential equation of second order, the variant space $I(u)<0$, the energy $E$ and the concavity function. In Section 3.1, 3.2, 3.3 and 3.4, we prove Theorem 1.1, Theorem 1.3, Corollary 1.5 and Corollary 1.6.

Next, we set a lemma with the fundamental statement for the ordinary differential inequality.

Here, we put

for $t\in[t_{0},T)$. We prove that the set $\mathcal{B}$ is invariant under our assumptions.