In this paper, we prove that the Cauchy problem for the following damped generalized IMBq equation,

$u_{tt}-u_\textit{\scriptsize xx}-u_\textit{\scriptsize xxtt}+\nu _2 u_\textit{\scriptsize xxt}=f(u)_\textit{\scriptsize xx}, x\in \mathbb {R}, t>0,$
utt−uxx−uxxtt+ν2uxxt=f(u)xx,x∈R,t>0, admits a unique global generalized solution in
$C^3([0,\infty );W^{m,p}(\mathbb {R})\cap L^\infty (\mathbb {R})\cap L^2 (\mathbb {R})) (1\break \leq p \leq \infty,m\ge 0)$
C3([0,∞);Wm,p(R)∩L∞(R)∩L2(R))(1≤p≤∞,m≥0) and a unique global classical solution in
$C^3([0,\infty );W^{m,p}(\mathbb {R})\cap L^\infty (\mathbb {R}) \cap L^2(\mathbb {R}) ) (m>2+\frac{1}{p})$
C3([0,∞);Wm,p(R)∩L∞(R)∩L2(R))(m>2+1p)⁠. Moreover, the blow up of the solution for the Cauchy problem of damped generalized IMBq equation is studied. We also prove that the Cauchy problem of the above-mentioned equation has a unique global generalized solution in
$C^2([0,\infty );H^s(\mathbb {R}) )(s > \frac12)$
C2([0,∞);Hs(R))(s>12) and a unique global classical solution in
$C^2([0,\infty );H^s(\mathbb {R}))(s>\frac{5}{2})$
C2([0,∞);Hs(R))(s>52)⁠, and discuss the blow up of the solution.

Topics
Functional analysis, Computational methods, Integral equations, Partial differential equations, Function space, Probability theory, Korteweg-de Vries equation, Surface waves
1.
E.
 
Arévalo
,
Y.
 
Gaididei
 and
F. G.
 
Mertens
, “
Soliton dynamics in damped and forced Boussinesq equations
,”
Eur. Phys. J. B
 
27
,
63
(
2002
).
https://doi.org/10.1140/epjb/e20020130
Google Scholar
Crossref
Search ADS
 
2.
G.
 
Chen
 and
S.
 
Wang
,
Proceedings of the Conference on Nonlinear Partial Differential Equations and Applications
(
World Scientific
,
Singapore
,
1998
).
3.
G.
 
Chen
 and
S.
 
Wang
, “
Existence and nonexistence of global solution for the generalized IMBq equation
,”
Nonlinear Anal. Theory, Methods Appl.
 
36
,
961
(
1999
).
https://doi.org/10.1016/S0362-546X(97)00710-4
Google Scholar
Crossref
Search ADS
 
4.
A.
 
Clarkson
,
R. J.
 
LeVeque
 and
R.
 
Saxton
, “
Solitary-wave interactions in elastic rods
,”
Stud. Appl. Math.
 
75
,
95
(
1986
).
Google Scholar
Crossref
Search ADS
 
5.
A.
 
Constantin
 and
L.
 
Molinet
, “
The initial value problem for a generalized Boussinesq equation
,”
Diff. Integral. Eq.
 
15
(
9
),
1061
(
2002
).
Google Scholar
 
6.
V. K.
 
Kalantarov
 and
O. A.
 
Ladyzhenskaya
, “
The occurrence of collapse for quasilinear equation of parabolic and hyperbolic types
,”
J. Sov. Math.
 
10
,
53
(
1978
).
https://doi.org/10.1007/BF01109723
Google Scholar
Crossref
Search ADS
 
7.
T.
 
Kato
 and
G.
 
Pouce
, “
Commutator estimates and the Euler and Navier-Stokes equations
,”
Commun. Pure Appl. Math.
 
41
,
891
(
1988
).
https://doi.org/10.1002/cpa.3160410704
Google Scholar
Crossref
Search ADS
 
8.
H. A.
 
Levine
, “
Some additional remark on the nonexistence of global solution to nonlinear wave equations
,”
SIAM J. Math. Anal.
 
5
,
138
(
1974
).
https://doi.org/10.1137/0505015
Google Scholar
Crossref
Search ADS
 
9.
T.
 
Li
 and
Y.
 
Chen
,
Global Classical Solutions for Nonlinear Evolution Equations
(
Longmans
 
New York
,
1992
), Vol.
45
.
Google Scholar
 
10.
Y.
 
Liu
, “
Existence and blow up of solutions of a nonlinear Pochhammer–Chree equation
,”
Indiana Univ. Math. J.
 
45
,
797
(
1996
).
Google Scholar
 
11.
V. G.
 
Makhankov
, “
Dynamics of classical solitons (in nonintegrable systems)
,”
Phys. Rep. Lett.
C
35
(
1
),
1
(
1978
).
https://doi.org/10.1016/0370-1573(78)90074-1
Google Scholar
Crossref
Search ADS
 
12.
E. M.
 
Stein
,
Singular Integrals and Differentiability Properties of Functions
(
Princeton University
,
New York
,
1970
).
Google Scholar
 
13.
S.
 
Wang
 and
G.
 
Chen
, “
The Cauchy problem for the generalized IMBq equation in Ws,p(Rn)
,”
J. Math. Anal. Appl.
 
266
,
38
(
2002
).
https://doi.org/10.1006/jmaa.2001.7670
Google Scholar
Crossref
Search ADS
 
14.
S.
 
Wang
 and
G.
 
Chen
, “
Small amplitude solutions of the generalized IMBq equation
,”
J. Math. Anal. Appl.
 
274
,
846
(
2002
).
https://doi.org/10.1016/S0022-247X(02)00401-8
Google Scholar
Crossref
Search ADS
 
© 2011 American Institute of Physics.
2011
American Institute of Physics
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