Asymptotic expansion associated with the Kirchhoff-Carrier- Love equation

By Vo Thi Tuyet Mai, Le Huu Ky Son, Tran Thi Kim Thoa
DOI: 10.37550/tdmu.EJS/2023.05.423

Abstract

In this paper, we consider the Dirichlet boundary problem for a nonlinear wave equation
of Kirchhoff-Carrier-Love type as follow


utt − B
?
∥u(t)∥2 , ∥ux(t)∥2
(uxx + uxxtt)
= f(x, t, u, ux, ut, uxt) +
Xp
i=1
εifi(x, t, u, ux, ut, uxt)
for 0 < x < 1, 0 < t < T,
u(0, t) = u(1, t) = 0,
u(x, 0) = ˜u0(x), ut(x, 0) = ˜u1(x),
(1)
where ˜u0, ˜u1, B, f, fi (i = 1, · · · , p) are given functions, ε1, · · · , εp are small parameters
and ∥u(t)∥2 =
Z 1
0
u2 (x, t) dx, ∥ux(t)∥2 =
Z 1
0
u2
x (x, t) dx. First, a declaration of the
existence and uniqueness of solutions provided by the linearly approximate technique
and the Faedo-Galerkin method is presented. Then, by using Taylor’s expansion for
the functions B, f, fi, i = 1, · · · , p, up to (N + 1)th order, we establish a high-order
asymptotic expansion of solutions in the small parameters ε1, · · · , εp.