The Self-interacting Scalar Field
Propagating in FLRW Model
of the Contracting Universe
Anahit Galstian and Karen Yagdjian
Abstract We present a condition on the self-interaction term that guaranties the
existence of the global-in-time solution of the Cauchy problem for the semilinear
Klein–Gordon equation in the FLRW model of the contracting universe. For the
equation with the Higgs potential, we give an estimate for the lifespan of solution.
1 Introduction and Statement of Results
In the present chapter, we prove the global-in-time existence of the solutions of
the Cauchy problem for the semilinear Klein–Gordon equation in the FLRW
(Friedmann–Lema
ˆ
itre–Robertson–Walker) space–time of the contracting universe
for the self-interacting scalar field.
The metric g in the FLRW space–time of the contracting universe in the
Lama
ˆ
itre–Robertson coordinates (see, e.g., [10]) is defined as follows, g
00
=
g
00
=−1, g
0j
= g
0j
= 0, g
ij
(x,t) = e
−2t
σ
ij
(x), i, j = 1, 2,...,n, where
∑
n
j =1
σ
ij
(x)σ
jk
(x) = δ
ik
, and δ
ij
is Kronecker’s delta. The metric σ
ij
(x) describes
the time slices. The covariant Klein–Gordon equation in that space–time in the
coordinates is
ψ
tt
−
e
2t
√
| det σ(x)|
n
i,j =1
∂
∂x
i
| det σ(x)|σ
ij
(x)
∂
∂x
j
ψ
− nψ
t
+ m
2
ψ = F(ψ).
(1.1)
It is obvious that the properties of this equation and of its solutions are not time
invertible. In the present chapter, we are interested in the Cauchy problem, which,
in fact, is not equivalent to the time backward problem for the equation with the
A. Galstian () · K. Yagdjian
School of Mathematical and Statistical Sciences, University of Texas RGV, Edinburg, TX, USA
e-mail: anahit.galstyan@utrgv.edu; karen.yagdjian@utrgv.edu
© Springer Nature Switzerland AG 2019
K.-O. Lindahl et al. (eds.), Analysis, Probability, Applications, and Computation,
Trends in Mathematics, https://doi.org/10.1007/978-3-030-04459-6_30
315