Author:
Karlsen K. H.,Rybalko Ya.
Abstract
AbstractThis study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class $$(m,n)\in C^{k}(\mathbb {R}) \cap W^{k,1}(\mathbb {R})$$
(
m
,
n
)
∈
C
k
(
R
)
∩
W
k
,
1
(
R
)
with $$k\in \mathbb {N}\cup \{0\}$$
k
∈
N
∪
{
0
}
. This system extends the celebrated Fokas–Olver–Rosenau–Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: $$\begin{aligned} \partial _t m(t,x)= \partial _x[m(t,x)(u(t,x)-\partial _xu(t,x)) (u(-t,-x)+\partial _x(u(-t,-x)))], \end{aligned}$$
∂
t
m
(
t
,
x
)
=
∂
x
[
m
(
t
,
x
)
(
u
(
t
,
x
)
-
∂
x
u
(
t
,
x
)
)
(
u
(
-
t
,
-
x
)
+
∂
x
(
u
(
-
t
,
-
x
)
)
)
]
,
where $$m(t,x)=\left( 1-\partial _{x}^2\right) u(t,x)$$
m
(
t
,
x
)
=
1
-
∂
x
2
u
(
t
,
x
)
. Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class $$C^k\cap W^{k,1}$$
C
k
∩
W
k
,
1
. Moreover, we derive criteria for blow-up of the local solution in this class.
Publisher
Springer Science and Business Media LLC