In this paper we consider a linear Cauchy viscoelastic problem. We show that, for compactly supported initial data and for not necessarily decreasing (oscillating) relaxation function, the decay of the first energy of solutions is polynomial. The proof relies on some appropriately chosen functionals. The combination of which satisfies a certain integral inequality.

1.
Aassila
,
M.
,
Cavalcanti
,
M. M.
, and
Soriano
,
J. A.
, “
Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain
,”
SIAM J. Control Optim.
38
,
1581
(
2000
).
2.
Berrimi
,
S.
and
Messaoudi
,
S. A.
, “
Existence and decay of solutions of a viscoelastic equation with a nonlinear source
,”
Nonlinear Anal.
64
,
2314
(
2006
).
3.
Cabanillas
,
E. L.
and
Munoz Rivera
,
J. E.
, “
Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomial decaying kernels
,”
Commun. Math. Phys.
177
,
583
(
1996
).
4.
Cavalcanti
,
M. M.
,
Domingos Cavalcanti
,
V. N.
, and
Ferreira
,
J.
, “
Existence and uniform decay for nonlinear viscoelastic equation with strong damping
,”
Math. Methods Appl. Sci.
24
,
1043
(
2001
).
5.
Cavalcanti
,
M. M.
,
Domingos Cavalcanti
,
V. N.
,
Prates Filho
,
J. S.
, and
Soriano
,
J. A.
, “
Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping
,”
Diff. Integral Eq.
14
,
85
(
2001
).
6.
Cavalcanti
,
M. M.
,
Domingos Cavalcanti
,
V. N.
, and
Santos
,
M. L.
, “
Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary
,”
Appl. Math. Comput.
150
,
439
(
2004
).
7.
Cavalcanti
,
M. M.
,
Domingos Cavalcanti
,
V. N.
, and
Soriano
,
J. A.
, “
Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping
,”
Electron. J. Differ. Equations
44
,
1
(
2002
).
8.
Cavalcanti
,
M. M.
and
Guesmia
,
A.
, “
General decay rates of solutions to a nonlinear wave equation with boundary conditions of memory type
,”
Diff. Integral Eq.
18
,
583
(
2005
).
9.
Cavalcanti
,
M. M.
and
Oquendo
,
H. P.
, “
Frictional versus viscoelastic damping in a semilinear wave equation
,”
SIAM J. Control Optim.
42
,
1310
(
2003
).
10.
Coleman
,
B. D.
and
Mizel
,
V. J.
, “
On the general theory of fading memory
,”
Arch. Ration. Mech. Anal.
29
,
18
(
1968
).
11.
Coleman
,
B. D.
and
Noll
,
W.
, “
Foundations of linear viscoelasticity
,”
Rev. Mod. Phys.
33
,
239
(
1961
).
12.
Dassios
,
G.
and
Zafiropoulos
,
F.
, “
Equipartition of energy in linearized 3-d viscoelasticity
,”
Q. Appl. Math.
48
,
715
(
1990
).
13.
Doherty
,
A.
and
Angela
,
M.
, “
Global solutions for a system Klein-Gordon equations with memory
,”
Boletim da Sociedade Paranaense de Matematica
21
,
127
(
2003
).
14.
Hrusa
,
W. J.
and
Nohel
,
J. A.
, “
The Cauchy problem in one-dimensional nonlinear viscoelasticity
,”
J. Differ. Equations
59
,
388
(
1985
).
15.
Kafini
,
M.
and
Messaoudi
,
S. A.
, “
A blow-up result for a viscoelastic system in RN
,”
Electron. J. Differ. Equations
113
,
1
(
2007
).
16.
Kafini
,
M.
and
Messaoudi
,
S. A.
, “
A blow-up result in a Cauchy viscoelastic problem
,”
Appl. Math. Lett.
21
,
549
(
2008
).
17.
Kafini
,
M.
and
Messaoudi
,
S. A.
, “
On the uniform decay in viscoelastic problems in Rn
,”
Appl. Math. Comput.
215
,
1161
(
2009
).
18.
Kirane
,
M.
and
Tatar
,
N. -e.
, “
Non-existence results for a semilinear hyperbolic problem with boundary condition of memory type
,”
Z. Anal. ihre Anwend.
19
,
453
(
2000
).
19.
Medjden
,
M.
and
Tatar
,
N. -e.
, “
Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel
,”
Appl. Math. Comput.
167
,
1221
(
2005
).
20.
Medjden
,
M.
and
Tatar
,
N. -e.
, “
On the wave equation with a temporal nonlocal term
,”
Dyn. Syst. Appl.
16
,
665
(
2007
).
21.
Messaoudi
,
S. A.
, “
Blow up and global existence in a nonlinear viscoelastic wave equation
,”
Math. Nachr.
260
,
58
(
2003
).
22.
Messaoudi
,
S. A.
, “
General decay of the solution energy in a viscoelastic equation with a nonlinear source
,”
Nonlinear Anal.
69
,
2589
(
2008
).
23.
Messaoudi
,
S. A.
, “
General decay of solutions of a viscoelastic equation
,”
J. Math. Anal. Appl.
341
,
1457
(
2008
).
24.
Munoz Rivera
,
J. E.
, “
Asymptotic behavior in linear viscoelasticity
,”
Q. Appl. Math.
52
,
628
(
1994
).
25.
Naso
,
M. G.
and
Munoz Rivera
,
J. E.
, “
On the decay of the energy for systems with memory and indefinite dissipation
,”
Asymptotic Anal.
49
,
189
(
2006
).
26.
Santos
,
M. L.
,
Raposo
,
C. A.
, and
Soares
,
U. R.
, “
On nonlinear coupled system with non-local boundary conditions
,”
Boletim da Sociedade Paranaense de Matematica
21
,
73
(
2003
).
You do not currently have access to this content.