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A general stability result for a nonlinear wave equation with infinite memory
Accepted Manuscript A general stability result for a nonlinear wave equation with infinite memory Salim A. Messaoudi, Muhammad M. Al-Gharabli PII: DOI: Reference:
S0893-9659(13)00169-9 http://dx.doi.org/10.1016/j.aml.2013.06.002 AML 4396
To appear in:
Applied Mathematics Letters
Received date: 8 May 2013 Accepted date: 11 June 2013 Please cite this article as: S.A. Messaoudi, M.M. Al-Gharabli, A general stability result for a nonlinear wave equation with infinite memory, Appl. Math. Lett. (2013), http://dx.doi.org/10.1016/j.aml.2013.06.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A general stability result for a nonlinear wave equation with infinite memory Salim A. Messaoudi(1) and Muhammad M. Al-Gharabli(2) King Fahd University of Petroleum and Minerals Department of Mathematics and Statistics Dhahran 31261, Saudi Arabia. (1) E-mail: [email protected] (2) E-mail: [email protected] Abstract In this paper we consider a nonlinear wave problem in the presence of an infinite-memory term and prove an explicit and general stability result. Our approach allows a wider class of kernels, from which those of exponential decay type, usually considered in the literature, are only special cases. Keywords and phrases: general decay, infinite memory, nonilinear, wave equation. AMS Classification: 35B37, 35L55, 74D05, 93D15, 93D20.
1
Introduction
In [1], Cavalcanti et al. discussed the problem Z t ρ g(t − s)∆u(s)ds − γ∆ut = 0, in Ω × (0, +∞) |ut | utt − ∆u − ∆utt + 0
u = 0 on Γ × (0, ∞) u(x, 0) = u0 (x), ut (x, 0) = u1 (x) in Ω,
(1.1)
where Ω is a bounded domain in Rn , n ≥ 1 with a smooth boundary Γ, ρ is a positive real number such that 0 < ρ ≤ 2/(n − 2) if n ≥ 3 or ρ > 0 if n = 1, 2 and g is a positive exponentially decaying function. They proved a global existence result when the constant γ ≥ 0 and an exponential decay result for the case γ > 0. Messaoudi and Tatar [2, 3] considered (1.1), for γ = 0, and showed that the energy of solution decay exponentially (resp. polynomially) if g decays exponentially (resp. polynomially). Later, Han and Wang [4] considered (1.1) for γ = 0 and with a relaxation function of more general decay type and established, similarly to the work of Messaoudi [5], a general decay result, from which the usual exponential and polynomial decay are only special cases. Recently, in [6], Messaoudi and Mustafa considered (1.1), for 1
relaxation functions satisfying a relation of the form g 0 (t) ≤ −H(g(t)), where H is a convex function satisfying some smoothness conditions. They established a general relation between the decay rate for the energy and that of the relaxation function g without imposing restrictive assumptions on the behavior of g at infinity. For more results related to problem (1.1), we refer the reader to Liu [7], [8] Our aim in this work is to investigate the following problem Z +∞ ρ |u | u − ∆u − ∆utt + g(s)∆u(t − s)ds = 0, in Ω × (0, +∞) t tt 0 u = 0 on Γ × (0, +∞) (1.2) u(x, −t) = u0 (x, t), ut (x, 0) = u1 (x) in Ω times(0, +∞),
for relaxation functions satisfying (2.3) below and obtain a general stability result for a wide class of kernels, from which those of exponential-decay type, usually considered in the literature, are only special cases.
2
Preliminaries
In order to state and prove our main result, we make the following hypotheses (H1) ρ is a constant satisfying 0<ρ≤ 0 < ρ,
2 , (n−2)
n≥3 n = 1, 2
(H2) g : IR+ → IR+ is a differentiable function satisfying Z +∞ g(s)ds = l > 0. g(0) > 0, 1−
(2.1)
(2.2)
0
(H3) There exists a nonincreasing differentiable function ξ : IR+ → IR+ such that g 0 (s) ≤ −ξ(s)g(s),
∀s ∈ IR+ .
(2.3)
For completeness we state, without proof, the existence result of [1]. Proposition 2.1. Let (u0 (., 0), u1 ) ∈ H01 (Ω) × H01 (Ω) be given. Assume that ( H1) ( H3) are satisfied, then problem (1.2) has a unique global (weak) solution u ∈ C 1 (IR+ ; H01 (Ω)). We introduce the ”modified” energy associated to problem (1.2) Z Z Z 1 l 1 1 ρ+2 2 E(t) = |ut | dx + |∇u| dx + |∇ut |2 dx + (g ◦ ∇u)(t), ρ+2 2 2 2 Ω
Ω
Ω
2
(2.4)
where, for any v ∈ L2 (Ω), (g ◦ v)(t) =
Z Z Ω
+∞
g(t − s)|v(t) − v(s)|2 dsdx.
0
A direct differentiation, using (1.2), leads to 1 E 0 (t) = (g 0 ◦ ∇u)(t) ≤ 0. 2
3
(2.5)
General decay
In this section we state and prove our main stability result. We will adopt two lemmas from [5] and [6] and state them without proofs. Our main result reads as follows: Theorem 3.1. Assume that ( H1) - ( H3) hold. Then for any (u0 , u1 ) ∈ H01 (Ω) × H01 (Ω) satisfying, for some m0 ≥ 0, k∇u0 (., s)k ≤ m0 ,
∀s > 0,
(3.1)
there exist constants γ0 ∈ (0, 1) and δ1 > 0 (depending only and continuously on ku0 k2H 1 ) such that, for all t ∈ IR+ and for all δ 0 ∈ (0, γ0 ], 0
E(t) ≤ δ1 (1 +
Z
t 1−δ0
(g(s))
ds)e
0
−δ0
Rt 0
ξ(s)ds
+ δ1
Z
+∞
g(s)ds
(3.2)
t
Examples. To illustrate our result, let us consider some examples. q 1. Let g(t) = de−(1+t) with 0 < q ≤ 1, and d > 0 small enough so that (2.2) and (2.3), with ξ(t) = q(1 + t)q−1 , hold. Then, (3.2) gives, for two positive constants c1 and c2 , q E(t) ≤ c1 e−c2 (1+t) , ∀t ∈ IR+
d with q > 1, and d > 0 small enough so that (2.2) and (2.3), (1 + t)q with ξ(t) = q(1 + t)−1 hold. Then, (3.2) gives, for two positive constants c1 and c2 , 2. Let g(t) =
E(t) ≤
c1 . ∀t ∈ IR+ . (t + 1)c2
Lemma 3.2. Under the assumptions ( H1) - ( H3), the functional Z Z 1 ρ Ψ(t) := |ut | ut udx + ∇u · ∇ut dx ρ+1 Ω
Ω
satisfies, along the solution of (1.2), the estimate Z Z Z l 1 1−l 2 2 0 Ψ (t) ≤ − |∇u| dx + |∇ut | dx + |ut |ρ+2 dx + (g ◦ ∇u)(t). (3.3) 2 ρ+1 2l Ω
Ω
Ω
3
Lemma 3.3. Under the assumptions ( H1) - ( H3), the functional Z +∞ Z |ut |ρ ut ∆ut − χ(t) := g(t − s) (u(t) − u(s)) dsdx ρ+1 0 Ω
satisfies, along the solution of (1.2) and for any δ1 , δ2 > 0, the estimate Z Z g0 2 0 2 χ (t) ≤ 1 + 2g0 δ1 |∇u| dx − |ut |ρ+2 dx ρ+1 Ω Ω g(0) Cp 1 (g ◦ ∇u)(t) − 1+ (g 0 ◦ ∇u)(t) +g0 2δ1 + 2δ1 4δ2 ρ+1 Z δ2 ρ + δ2 + c (2 E(0)) − g0 |∇ut |2 dx. (3.4) ρ+1 Ω
where g0 =
Z
∞
g(s)ds, c is a positive constant and Cp is the Poincar´e inequality.
0
Lemma 3.4. Assume that ( H1) - ( H3) hold. Then there exist constants ε, α1 , α2 , M > 0 such that the functional L = M E + εΨ + χ satisfies, for all t ∈ IR+ , 0
L (t) ≤ −α1 E(t) + α2
Z
0
+∞
g(s)||∇u(t − s) − ∇u(t)||2 ds.
(3.5)
Proof. By using (2.5), (3.3), (3.4), we easily see that Z g(0) Cp g0 − ε M 0 L (t) ≤ − 1+ (g ◦ ∇u)(t) − |ut |ρ+2 dx (3.6) 2 4δ2 ρ+1 ρ+1 Ω Z Z l 2 ρ 2 − ε − 1 + 2g0 δ1 |∇u| dx − [g0 − ε − δ2 − cδ2 (2 E(0)) ] |∇ut |2 dx 2 Ω Ω ε 1 +g0 + 2δ1 + (g ◦ ∇u)(t) 2l 2δ1 0
We now choose our constant carefully. First, we pick ε < g0 , then δ1 and δ2 small enough that g0 − ε > 0,
ε
1 − g0 − 1 + 2g02 δ1 > 0, 2
g0 − ε − δ2 − cδ2 (2 E(0))ρ > 0.
Then we take M sufficiently large so that M g(0) Cp − 1+ ≥ 0. 2 4δ2 ρ+1 4
Therefore, (3.6) reduces to (3.5) for two positive constants α1 and α2 . Lemma 3.5. Assume that (H1)-(H3) and (3.1) are satisfied. Then there exist positive constants β1 and β2 such that, for all t ∈ IR+ , Z +∞ 0 0 ξ(t)L (t) + β1 E (t) ≤ −α1 ξ(t)E(t) + β2 ξ(t) g(s)ds (3.7) t
Proof. Using (2.3), (2.5) and the fact that ξ and g are nonincreasing, we get Z t Z t 2 g(s)||∇u(t − s) − ∇u(t)|| ds ≤ − g 0 (s)||∇u(t − s) − ∇u(t)||2 ds ξ(t) 0
0
≤ −2E 0 (t), ∀t ∈ IR+
(3.8)
On the other hand, (2.4) and the fact that E is nonincreasing imply that k∇u(t − s) − ∇u(t)k2 ≤ 2k∇u(t − s)k2 + 2k∇u(t)k2 ≤ 4 sup k∇u(s)k2 + 2 sup k∇u(τ )k2 s>0
τ <0
4 ≤ E(0) + 2 sup k∇u0 (τ )k2 , l τ >0
∀t, s ∈ IR+
Hence, combination with (3.1) yields, for all t ∈ IR+ , Z +∞ Z +∞ 4 2 2 g(s)k∇u(t − s) − ∇u(t)k ds ≤ ξ(t) E(0) + 2m0 ξ(t) g(s)ds l t t
(3.9)
Finally, multiplying (3.5) by ξ(t) and combining with (3.8) and (3.9), we get (3.7). Proof of Theorem 3.1. Now we consider the functionals Z +∞ g(s)ds (3.10) F = ξL + β1 E and h(t) = ξ(t) t
By using the fact that F ∼ E and ξ is nonincreasing, (3.7) gives F 0 (t) ≤ −γ0 ξ(t)F (t) + β2 h(t),
∀t ∈ IR+
for some γ0 > 0. This last inequality remains true for any δ0 ∈ (0, γ0 ]; that is F 0 (t) ≤ −δ0 ξ(t)F (t) + β2 h(t),
∀t ∈ IR+ .
Therefore, direct integrating leads to F (T ) ≤ e
−δ0
RT 0
ξ(s)ds
F (0) + β2
Z
T
eδ0
0
and the fact that F ∼ E gives
RT 1 E(T ) ≤ e−δ0 0 ξ(s)ds F (0) + β2 β1 5
Z
0
Rt 0
T
eδ0
ξ(s)ds
Rt 0
h(t)dt
ξ(s)ds
h(t)dt
(3.11)
By noting that ZT 0
δ0
e
Rt 0
ξ(s)ds
1 δ0 R t ξ(s)ds 0 h(t)dt = e 0 δ0
Z
+∞
g(s)ds,
t
∀t ∈ IR+ ,
then integration by parts leads to Z T Z +∞ Z Z T R R 1 δ0 R T ξ(s)ds +∞ δ0 0t ξ(s)ds δ0 0t ξ(s)ds 0 g(t)dt . e g(s)ds+ g(s)ds− h(t)dt = e e δ0 0 0 T 0 Consequently, combining with (3.11), Z
1 β2 E(T ) ≤ F (0) + β1 δ0
T
e
δ0
0
Rt 0
ξ(s)ds
Z +∞ R β2 −δ0 0T ξ(s)ds + g(t)dt e g(s)ds (3.12) β 1 δ0 T
Also, (2.3) implies that
Rt
Consequently, we have e Z
0
T
0
e
δ0
Rt
e
0
ξ(s)ds Rt 0
ξ(s)ds
0 g(t) ≤ 0,
∀t ∈ IR+ .
g(t) ≤ g(0) and
ξ(s)ds
δ0
g(t)dt ≤ (g(0))
Z
T
(g(t))1−δ0 dt
(3.13)
0
Finally, we obtain (3.2) by combining (2.4), (3.12) and (3.13). Acknowledgment. The authors thank KFUPM for its continuous support. This work has been funded by KFUPM under Project # FT121007.
4
References [1] M.M. Cavalcanti, V.N. Domingos Cavalcanti, and J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Meth. Appl. Sci. 24 (2001), 1043 - 1053. [2] S.A. Messaoudi and N-E. Tatar, Exponential and polynomial decay for a quasilinear viscoelastic equation, Nonlinear Anal. TMA 68 (2007), 785-793. [3] S.A. Messaoudi and N-E. Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Meth. Appl. Sci. 30 (2007), 665-680. [4] X. Han and M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping, Math. Meth. Appl. Sci. 32 # 3 (2009), 346-358. [5] S.A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl. 341 (2008), 1457-1467. 6
[6] S.A. Messaoudi and M. I.Mustafa, General stability result for a quasilinear wave equation with memory, Nonlinear Analysis: Real World Applications 14 (2013) 1854-1864 [7] W.J. Liu, General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms, J. Math. Phys. 50 (11): Art. No. 113506 NOV 2009. [8] W.J. Liu, General decay and blow up of solution for a quasilinear viscoelastic equation with a nonlinear source, Nonlinear Anal. 73 (2010), 1890-1904.
7
S0893-9659(13)00169-9 http://dx.doi.org/10.1016/j.aml.2013.06.002 AML 4396
To appear in:
Applied Mathematics Letters
Received date: 8 May 2013 Accepted date: 11 June 2013 Please cite this article as: S.A. Messaoudi, M.M. Al-Gharabli, A general stability result for a nonlinear wave equation with infinite memory, Appl. Math. Lett. (2013), http://dx.doi.org/10.1016/j.aml.2013.06.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A general stability result for a nonlinear wave equation with infinite memory Salim A. Messaoudi(1) and Muhammad M. Al-Gharabli(2) King Fahd University of Petroleum and Minerals Department of Mathematics and Statistics Dhahran 31261, Saudi Arabia. (1) E-mail: [email protected] (2) E-mail: [email protected] Abstract In this paper we consider a nonlinear wave problem in the presence of an infinite-memory term and prove an explicit and general stability result. Our approach allows a wider class of kernels, from which those of exponential decay type, usually considered in the literature, are only special cases. Keywords and phrases: general decay, infinite memory, nonilinear, wave equation. AMS Classification: 35B37, 35L55, 74D05, 93D15, 93D20.
1
Introduction
In [1], Cavalcanti et al. discussed the problem Z t ρ g(t − s)∆u(s)ds − γ∆ut = 0, in Ω × (0, +∞) |ut | utt − ∆u − ∆utt + 0
u = 0 on Γ × (0, ∞) u(x, 0) = u0 (x), ut (x, 0) = u1 (x) in Ω,
(1.1)
where Ω is a bounded domain in Rn , n ≥ 1 with a smooth boundary Γ, ρ is a positive real number such that 0 < ρ ≤ 2/(n − 2) if n ≥ 3 or ρ > 0 if n = 1, 2 and g is a positive exponentially decaying function. They proved a global existence result when the constant γ ≥ 0 and an exponential decay result for the case γ > 0. Messaoudi and Tatar [2, 3] considered (1.1), for γ = 0, and showed that the energy of solution decay exponentially (resp. polynomially) if g decays exponentially (resp. polynomially). Later, Han and Wang [4] considered (1.1) for γ = 0 and with a relaxation function of more general decay type and established, similarly to the work of Messaoudi [5], a general decay result, from which the usual exponential and polynomial decay are only special cases. Recently, in [6], Messaoudi and Mustafa considered (1.1), for 1
relaxation functions satisfying a relation of the form g 0 (t) ≤ −H(g(t)), where H is a convex function satisfying some smoothness conditions. They established a general relation between the decay rate for the energy and that of the relaxation function g without imposing restrictive assumptions on the behavior of g at infinity. For more results related to problem (1.1), we refer the reader to Liu [7], [8] Our aim in this work is to investigate the following problem Z +∞ ρ |u | u − ∆u − ∆utt + g(s)∆u(t − s)ds = 0, in Ω × (0, +∞) t tt 0 u = 0 on Γ × (0, +∞) (1.2) u(x, −t) = u0 (x, t), ut (x, 0) = u1 (x) in Ω times(0, +∞),
for relaxation functions satisfying (2.3) below and obtain a general stability result for a wide class of kernels, from which those of exponential-decay type, usually considered in the literature, are only special cases.
2
Preliminaries
In order to state and prove our main result, we make the following hypotheses (H1) ρ is a constant satisfying 0<ρ≤ 0 < ρ,
2 , (n−2)
n≥3 n = 1, 2
(H2) g : IR+ → IR+ is a differentiable function satisfying Z +∞ g(s)ds = l > 0. g(0) > 0, 1−
(2.1)
(2.2)
0
(H3) There exists a nonincreasing differentiable function ξ : IR+ → IR+ such that g 0 (s) ≤ −ξ(s)g(s),
∀s ∈ IR+ .
(2.3)
For completeness we state, without proof, the existence result of [1]. Proposition 2.1. Let (u0 (., 0), u1 ) ∈ H01 (Ω) × H01 (Ω) be given. Assume that ( H1) ( H3) are satisfied, then problem (1.2) has a unique global (weak) solution u ∈ C 1 (IR+ ; H01 (Ω)). We introduce the ”modified” energy associated to problem (1.2) Z Z Z 1 l 1 1 ρ+2 2 E(t) = |ut | dx + |∇u| dx + |∇ut |2 dx + (g ◦ ∇u)(t), ρ+2 2 2 2 Ω
Ω
Ω
2
(2.4)
where, for any v ∈ L2 (Ω), (g ◦ v)(t) =
Z Z Ω
+∞
g(t − s)|v(t) − v(s)|2 dsdx.
0
A direct differentiation, using (1.2), leads to 1 E 0 (t) = (g 0 ◦ ∇u)(t) ≤ 0. 2
3
(2.5)
General decay
In this section we state and prove our main stability result. We will adopt two lemmas from [5] and [6] and state them without proofs. Our main result reads as follows: Theorem 3.1. Assume that ( H1) - ( H3) hold. Then for any (u0 , u1 ) ∈ H01 (Ω) × H01 (Ω) satisfying, for some m0 ≥ 0, k∇u0 (., s)k ≤ m0 ,
∀s > 0,
(3.1)
there exist constants γ0 ∈ (0, 1) and δ1 > 0 (depending only and continuously on ku0 k2H 1 ) such that, for all t ∈ IR+ and for all δ 0 ∈ (0, γ0 ], 0
E(t) ≤ δ1 (1 +
Z
t 1−δ0
(g(s))
ds)e
0
−δ0
Rt 0
ξ(s)ds
+ δ1
Z
+∞
g(s)ds
(3.2)
t
Examples. To illustrate our result, let us consider some examples. q 1. Let g(t) = de−(1+t) with 0 < q ≤ 1, and d > 0 small enough so that (2.2) and (2.3), with ξ(t) = q(1 + t)q−1 , hold. Then, (3.2) gives, for two positive constants c1 and c2 , q E(t) ≤ c1 e−c2 (1+t) , ∀t ∈ IR+
d with q > 1, and d > 0 small enough so that (2.2) and (2.3), (1 + t)q with ξ(t) = q(1 + t)−1 hold. Then, (3.2) gives, for two positive constants c1 and c2 , 2. Let g(t) =
E(t) ≤
c1 . ∀t ∈ IR+ . (t + 1)c2
Lemma 3.2. Under the assumptions ( H1) - ( H3), the functional Z Z 1 ρ Ψ(t) := |ut | ut udx + ∇u · ∇ut dx ρ+1 Ω
Ω
satisfies, along the solution of (1.2), the estimate Z Z Z l 1 1−l 2 2 0 Ψ (t) ≤ − |∇u| dx + |∇ut | dx + |ut |ρ+2 dx + (g ◦ ∇u)(t). (3.3) 2 ρ+1 2l Ω
Ω
Ω
3
Lemma 3.3. Under the assumptions ( H1) - ( H3), the functional Z +∞ Z |ut |ρ ut ∆ut − χ(t) := g(t − s) (u(t) − u(s)) dsdx ρ+1 0 Ω
satisfies, along the solution of (1.2) and for any δ1 , δ2 > 0, the estimate Z Z g0 2 0 2 χ (t) ≤ 1 + 2g0 δ1 |∇u| dx − |ut |ρ+2 dx ρ+1 Ω Ω g(0) Cp 1 (g ◦ ∇u)(t) − 1+ (g 0 ◦ ∇u)(t) +g0 2δ1 + 2δ1 4δ2 ρ+1 Z δ2 ρ + δ2 + c (2 E(0)) − g0 |∇ut |2 dx. (3.4) ρ+1 Ω
where g0 =
Z
∞
g(s)ds, c is a positive constant and Cp is the Poincar´e inequality.
0
Lemma 3.4. Assume that ( H1) - ( H3) hold. Then there exist constants ε, α1 , α2 , M > 0 such that the functional L = M E + εΨ + χ satisfies, for all t ∈ IR+ , 0
L (t) ≤ −α1 E(t) + α2
Z
0
+∞
g(s)||∇u(t − s) − ∇u(t)||2 ds.
(3.5)
Proof. By using (2.5), (3.3), (3.4), we easily see that Z g(0) Cp g0 − ε M 0 L (t) ≤ − 1+ (g ◦ ∇u)(t) − |ut |ρ+2 dx (3.6) 2 4δ2 ρ+1 ρ+1 Ω Z Z l 2 ρ 2 − ε − 1 + 2g0 δ1 |∇u| dx − [g0 − ε − δ2 − cδ2 (2 E(0)) ] |∇ut |2 dx 2 Ω Ω ε 1 +g0 + 2δ1 + (g ◦ ∇u)(t) 2l 2δ1 0
We now choose our constant carefully. First, we pick ε < g0 , then δ1 and δ2 small enough that g0 − ε > 0,
ε
1 − g0 − 1 + 2g02 δ1 > 0, 2
g0 − ε − δ2 − cδ2 (2 E(0))ρ > 0.
Then we take M sufficiently large so that M g(0) Cp − 1+ ≥ 0. 2 4δ2 ρ+1 4
Therefore, (3.6) reduces to (3.5) for two positive constants α1 and α2 . Lemma 3.5. Assume that (H1)-(H3) and (3.1) are satisfied. Then there exist positive constants β1 and β2 such that, for all t ∈ IR+ , Z +∞ 0 0 ξ(t)L (t) + β1 E (t) ≤ −α1 ξ(t)E(t) + β2 ξ(t) g(s)ds (3.7) t
Proof. Using (2.3), (2.5) and the fact that ξ and g are nonincreasing, we get Z t Z t 2 g(s)||∇u(t − s) − ∇u(t)|| ds ≤ − g 0 (s)||∇u(t − s) − ∇u(t)||2 ds ξ(t) 0
0
≤ −2E 0 (t), ∀t ∈ IR+
(3.8)
On the other hand, (2.4) and the fact that E is nonincreasing imply that k∇u(t − s) − ∇u(t)k2 ≤ 2k∇u(t − s)k2 + 2k∇u(t)k2 ≤ 4 sup k∇u(s)k2 + 2 sup k∇u(τ )k2 s>0
τ <0
4 ≤ E(0) + 2 sup k∇u0 (τ )k2 , l τ >0
∀t, s ∈ IR+
Hence, combination with (3.1) yields, for all t ∈ IR+ , Z +∞ Z +∞ 4 2 2 g(s)k∇u(t − s) − ∇u(t)k ds ≤ ξ(t) E(0) + 2m0 ξ(t) g(s)ds l t t
(3.9)
Finally, multiplying (3.5) by ξ(t) and combining with (3.8) and (3.9), we get (3.7). Proof of Theorem 3.1. Now we consider the functionals Z +∞ g(s)ds (3.10) F = ξL + β1 E and h(t) = ξ(t) t
By using the fact that F ∼ E and ξ is nonincreasing, (3.7) gives F 0 (t) ≤ −γ0 ξ(t)F (t) + β2 h(t),
∀t ∈ IR+
for some γ0 > 0. This last inequality remains true for any δ0 ∈ (0, γ0 ]; that is F 0 (t) ≤ −δ0 ξ(t)F (t) + β2 h(t),
∀t ∈ IR+ .
Therefore, direct integrating leads to F (T ) ≤ e
−δ0
RT 0
ξ(s)ds
F (0) + β2
Z
T
eδ0
0
and the fact that F ∼ E gives
RT 1 E(T ) ≤ e−δ0 0 ξ(s)ds F (0) + β2 β1 5
Z
0
Rt 0
T
eδ0
ξ(s)ds
Rt 0
h(t)dt
ξ(s)ds
h(t)dt
(3.11)
By noting that ZT 0
δ0
e
Rt 0
ξ(s)ds
1 δ0 R t ξ(s)ds 0 h(t)dt = e 0 δ0
Z
+∞
g(s)ds,
t
∀t ∈ IR+ ,
then integration by parts leads to Z T Z +∞ Z Z T R R 1 δ0 R T ξ(s)ds +∞ δ0 0t ξ(s)ds δ0 0t ξ(s)ds 0 g(t)dt . e g(s)ds+ g(s)ds− h(t)dt = e e δ0 0 0 T 0 Consequently, combining with (3.11), Z
1 β2 E(T ) ≤ F (0) + β1 δ0
T
e
δ0
0
Rt 0
ξ(s)ds
Z +∞ R β2 −δ0 0T ξ(s)ds + g(t)dt e g(s)ds (3.12) β 1 δ0 T
Also, (2.3) implies that
Rt
Consequently, we have e Z
0
T
0
e
δ0
Rt
e
0
ξ(s)ds Rt 0
ξ(s)ds
0 g(t) ≤ 0,
∀t ∈ IR+ .
g(t) ≤ g(0) and
ξ(s)ds
δ0
g(t)dt ≤ (g(0))
Z
T
(g(t))1−δ0 dt
(3.13)
0
Finally, we obtain (3.2) by combining (2.4), (3.12) and (3.13). Acknowledgment. The authors thank KFUPM for its continuous support. This work has been funded by KFUPM under Project # FT121007.
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[6] S.A. Messaoudi and M. I.Mustafa, General stability result for a quasilinear wave equation with memory, Nonlinear Analysis: Real World Applications 14 (2013) 1854-1864 [7] W.J. Liu, General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms, J. Math. Phys. 50 (11): Art. No. 113506 NOV 2009. [8] W.J. Liu, General decay and blow up of solution for a quasilinear viscoelastic equation with a nonlinear source, Nonlinear Anal. 73 (2010), 1890-1904.
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