In this paper, we investigate the influence of dissipation on decay properties of the solutions for a quasilinear beam equation with nonlinear weak dissipation.

1.
F.
Alabau-Boussouira
, “
Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems
,”
Appl. Math. Optim.
51
,
61
(
2005
).
2.
V. I.
Arnold
,
Mathematical Methods of Classical Mechanics
(
Springer
,
New York
,
1989
).
3.
J. M.
Ball
, “
Stability theory for an extensible beam
,”
J. Differ. Equations
14
,
399
(
1973
).
4.
P.
Biler
, “
Remark on the decay for generalized damped string and beam equations
,”
Nonlinear Anal.
10
,
839
(
1986
).
5.
A.
Benaissa
and
A.
Guesmia
, “
Energy decay for wave equations of ϕ-Laplacian type with weakly nonlinear dissipation
,”
Electron. J. Differ. Equations
2008
,
1
(
2008
).
6.
E. H.
de Brito
, “
Decay estimates for the generalized damped extensible string and beam equations
,”
Nonlinear Anal.
8
,
1489
(
1984
).
7.
D.
Burgreen
, “
Free vibrations of a pin-ended column with constant distance between pin ends
,”
ASME Trans. J. Appl. Mech.
18
,
135
(
1951
).
8.
M. M.
Cavalcanti
,
V. N. D.
Cavalcanti
, and
I.
Lasiecka
, “
Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction
,”
J. Differ. Equations
236
,
407
(
2007
).
9.
M. M.
Cavalcanti
,
V. N.
Domingos Cavalcanti
, and
P.
Martinez
, “
General decay rate estimates for viscoelastic dissipative systems
,”
Nonlinear Anal. Theory, Methods Appl.
68
,
177
(
2008
).
10.
R. W.
Dickey
, “
Free vibrations and dynamic buckling of the extensible beam
,”
J. Math. Anal. Appl.
29
,
443
(
1970
).
11.
J. G.
Eisley
, “
Nonlinear vibrations of beams and rectangular plates
,”
Z. Angew. Math. Phys.
15
,
167
(
1964
).
12.
A.
Guesmia
and
S. A.
Messaoudi
, “
General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping
,”
Math. Methods Appl. Sci.
32
,
2102
(
2009
).
13.
A.
Haraux
and
E.
Zuazua
, “
Decay estimates for some semilinear damped hyperbolic problems
,”
Arch. Ration. Mech. Anal.
100
,
191
(
1988
).
14.
R.
Ikehata
and
Y. K.
Inoue
, “
Global existence of weak solutions for two-dimensional semilinear wave equations with strong damping in an exterior domain
,”
Nonlinear Anal. Theory, Methods Appl.
68
,
154
(
2008
).
15.
V.
Komonik
,
Exact Controllability and Stabilization: The Multiplier Method
(
Wiley
,
New York
/
Masson
,
Paris
,
1994
).
16.
I.
Lasiecka
and
D.
Tataru
, “
Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping
,”
Diff. Integral Eq.
6
,
507
(
1993
).
17.
I.
Lasiecka
and
R.
Triggiani
, “
Well-posedness and sharp uniform decay rates at the L2(Ω)-level of the Schrödinger equation with nonlinear boundary dissipation
,”
J. Evol. Equ.
6
,
485
(
2006
).
18.
J. L.
Lions
and
E.
Magenes
,
Non-Homogeneous Boundary Value Problem and Applications
(
Springer-Verlag
,
New York
,
1972
), Vol.
I
.
19.
P.
Martinez
, “
A new method to obtain decay rate estimates for dissipative systems
,”
ESAIM: COCV
4
,
419
(
1999
).
20.
L. A.
Medeiros
, “
On a new class of nonlinear wave equations
,”
J. Math. Anal. Appl.
69
,
252
(
1979
).
21.
S. D. B.
Menezes
,
E. A.
Oliveira
,
D. C.
Pereira
, and
J.
Ferreira
, “
Existence, uniqueness and uniform decay for the nonlinear beam degenerate equation with weak damping
,”
Appl. Math. Comput.
154
,
555
(
2004
).
22.
G. P.
Menzala
, “
On classical solutions of a quasilinear hyperbolic equation
,”
Nonlinear Anal.
3
,
613
(
1979
).
23.
M.
Nakao
, “
Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term
,”
J. Math. Anal. Appl.
58
,
336
(
1977
).
24.
M.
de Lacerda Oliveira
and
O. A.
Lima
, “
Exponential decay of the solutions of the beam system
,”
Nonlinear Anal.
42
,
1271
(
2000
).
25.
S.
Woinowsky-Krieger
, “
The effect of axial force on the vibration of hinged bars
,”
ASME Trans. J. Appl. Mech.
17
,
35
(
1950
).
26.
Y.
Zhijian
, “
Cauchy problem for quasi-linear wave equations with viscous damping
,”
J. Math. Anal. Appl.
320
,
859
(
2006
).
27.
E.
Zuazua
, “
Stability and decay for a class of nonlinear hyperbolic problems
,”
Asymptotic Anal.
1
,
161
(
1988
).
You do not currently have access to this content.