Abstract
Abstract
Nonlinear partial differential equations appear in many domains of physics, and we study here a typical equation which one finds in effective field theories originated from cosmological studies. In particular, we are interested in the equation
∂
t
2
u
(
x
,
t
)
=
α
(
∂
x
u
(
x
,
t
)
)
2
+
β
∂
x
2
u
(
x
,
t
)
in
1
+
1
dimensions. It has been known for quite some time that solutions to this equation diverge in finite time, when α > 0. We study the nature of this divergence as a function of the parameters α > 0 and
β
⩾
0
. The divergence does not disappear even when β is very large contrary to what one might believe (note that since we consider fixed initial data, α and β cannot be scaled away). But it will take longer to appear as β increases when α is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to
3
+
1
dimensions.
Funder
National Centres of Competence in Research SwissMAP
ERC Advanced Grant
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics