Abstract
In this paper, we consider the nonlinear viscoelastic equation
Mathematical Subject Classification (2010): 35B35, 35B40, 35B60
Keywords:
Viscoelastic equation; Kernel function; Exponential decay; Polynomial decay1 Introduction
It is well known that viscoelastic materials have memory effects. These properties are due to the mechanical response influenced by the history of the materials themselves. As these materials have a wide application in the natural sciences, their dynamics are of great importance and interest. From the mathematical point of view, their memory effects are modeled by an integrodifferential equations. Hence, questions related to the behavior of the solutions for the PDE system have attracted considerable attention in recent years. Many authors have focused on this problem for the last two decades and several results concerning existence, decay and blowup have been obtained, see [128] and the reference therein.
In [3], Cavalcanti et al. studied the following problem
where Ω ⊂ R^{N}, N ≥ 1, is a bounded domain with a smooth boundary ∂Ω, γ ≥ 0,
and
for all t ≥ 0 and some positive constants ξ_{1 }and ξ_{2}. Later, this result was extended by Messaoudi and Tatar [15] to a situation where a nonlinear source term is competing with the dissipation terms induced by both the viscoelasticity and the viscosity. Recently Messaoudi and Tatar [14] studied problem (1.1) for the case of γ = 0, they improved the result in [3] by showing that the solution goes to zero with an exponential or polynomial rate, depending on the decay rate of the relaxation function g.
The assumptions (1.2) and (1.3), on g, are frequently encountered in the linear case (ρ = 0), see [1,2,46,13,22,23,2931]. Lately, these conditions have been weakened by some researchers. For instance, instead of (1.3) Furati and Tatar [8] required the functions e^{αt }g(t) and e^{αt}g'(t) to have sufficiently small L^{1}norm on (0, ∞) for some α > 0 and they can also have an exponential decay of solutions. In particular, they do not impose a rate of decreasingness for g. Later on Messaoudi and Tatar [21] improved this result further by removing the condition on g'. They established an exponential decay under the conditions g'(t) ≤ 0 and e^{αt }g(t) ∈ L^{1}(0, ∞) for some large α > 0. This last condition was shown to be necessary condition for exponential decay [7]. More recently Tatar [25] investigated the asymptotic behavior to problem (1.1) with ρ = γ = 0 when h(t)g(t) ∈ L^{1}(0, ∞) for some nonnegative function h(t). He generalized earlier works to an arbitrary decay not necessary of exponential or polynomial rate.
Motivated by previous works [21,25], in this paper, we consider the initial boundary value problem for the following nonlinear viscoelastic equation:
with initial conditions
and boundary condition
where Ω ⊂ R^{N}, N ≥ 1, is a bounded domain with a smooth boundary ∂Ω. Here ρ, p > 0 and g represents the kernel of the memory term, with conditions to be stated later [see assumption (A1)(A3)].
We intend to study the arbitrary decay result for problem (1.4)(1.6) under the weaker
assumption on g, which is not necessarily decaying in an exponential or polynomial fashion. Indeed,
our result will be established under the conditions g'(t) ≤ 0 and
The content of this paper is organized as follows. In Section 2, we give some lemmas and assumptions which will be used later, and we mention the local existence result in Theorem 2.2. In Section 3, we establish the statement and proof of our result related to the arbitrary decay.
2 Preliminary results
In this section, we give some assumptions and lemmas which will be used throughout
this work. We use the standard Lebesgue space L^{p}(Ω) and Sobolev space
Lemma 2.1. (SobolevPoincaré inequality) Let
holds with the optimal positive constant c_{s}, where  · _{p }denotes the norm of L^{p}(Ω).
Assume that ρ satisfies
With regards to the relaxation function g(t), we assume that it verifies
(A1) g(t) ≥ 0, for all t ≥ 0, is a continuous function satisfying
(A2) g'(t) ≤ 0 for almost all t > 0.
(A3) There exists a positive nondecreasing function ξ(t): [0, ∞) → (0, ∞) such that
Now, we state, without a proof, the existence result of the problem (1.4)(1.6) which can be established by FaedoGalerkin methods, we refer the reader to [3,5].
Theorem 2.2. Suppose that (2.1) and (A1) hold, and that
Next, we introduce the modified energy functional for problem (1.4)(1.6)
where
Lemma 2.3. Let u be the solution of (1.4)(1.6), then the modified energy E(t) satisfies
Proof. Multiplying Eq. (1.4) by u_{t }and integrating it over Ω, then using integration by parts and the assumption (A1)(A2), we obtain (2.6).
Remark. It follows from Lemma 2.3 that the energy is uniformly bounded by E(0) and decreasing in t. Besides, from the definition of E(t) and (2, 2), we note that
3 Decay of the solution energy
In this section, we shall state and prove our main result. For this purpose, we first define the functional
where λ_{i }are positive constants, i = 1, 2, 3 to be specified later and
here
Remark. This functional was first introduced by Tatar [25] for the case of ρ = 0 and without imposing the dispersion term and forcing term as far as (1.4) is concerned.
The following Lemma tells us that L(t) and E(t) + Φ_{3}(t) are equivalent.
Lemma 3.1. There exists two positive constants β_{1 }and β_{2 }such that the relation
holds for all t ≥ 0 and λ_{i }small, i = 1, 2.
Proof. By Hölder inequality Young's inequality Lemma 2.1, (2.7) and (2.2), we deduce that
and
where
and
where
and again from the definition of E(t), there exist two positive constants β_{1 }and β_{2 }such that
To obtain a better estimate for
Lemma 3.2. For t ≥ 0, we have
Proof. Straightforward computations yield this identity.
Now, we are ready to state and prove our result. First, we introduce the following
notations as in [24,25]. For every measurable set A ⊂ R^{+}, we define the probability measure
The flatness set and the flatness rate of g are defined by
and
Before proceeding, we note that there exists t_{0 }> 0 such that
since g is nonnegative and continuous.
Theorem 3.3. Let
Then the solution energy of (1.4)(1.6) satisfies
where μ and K are positive constants.
Proof. In order to obtain the decay result of E(t), it suffices to prove that of L(t). To this end, we need to estimate the derivative of L(t). It follows from (3.2) and Eq. (1.4) that
which together with the identity (3.6) and (2.2) gives
Next, we would like to estimate
We now estimate the first two terms on the righthand side of (3.11) as in [25].
Indeed, for all measure set A and F such that A = R^{+ } F, we have
To simplify notations, we denote
Using Hölder inequality Young's inequality and (2.2), we see that, for δ_{1 }> 0,
and
Thus, from the definition of
The second term on the righthand side of (3.11) can be estimated as follows (see [25]), for δ_{2 }> 0,
Using Hölder inequality Young's inequality and (A2) to deal with the fifth term, for δ_{3 }> 0,
Exploiting Hölder inequality Young's inequality Lemma 2.1 and (A2) to estimate the sixth term, for δ_{4 }> 0,
For the last term, thanks to Hölder inequality Young's inequality Lemma 2.1, (2.7), (2.2) and (3.8), we have, for δ_{5 }> 0,
where
Further, taking a derivative of Φ_{3}(t), using the fact that
Hence, we conclude from (2.6), (3.10), (3.18) and (3.19) that for any t ≥ t_{0 }> 0,
For
and observe that
where F_{g }is given in (3.7) and N_{g }is the null set where g' is not defined. In addition, denoting F_{n }= R^{+ } A_{n}, then
because A_{n }are increasingly nested. Thus, choosing A = A_{n}, F = F_{n }and λ_{1 }= (g_{* } ε) λ_{2 }for some ε > 0 in (3.20), we obtain
At this point, we take
then (3.21) becomes
For ε, δ_{2 }small enough and large value of n and t_{0}, we see that if
then
and
where
and
Note that α > 0 and 0 < δ < 1 due to
and
this is possible because of
Hence, from the definition of E(t) by (2.4), we have, for all t ≥ t_{0},
for some positive constant c_{4}. As η(t) is decreasing, we have η(t) ≤ c_{4 }after some t_{* }≥ t_{0}. Hence, with the help of the right hand side inequality in (3.5), we find
for some positive constant c_{5 }> 0. An integration of (3.23) over (t_{*}, t) gives
Then using the left hand side inequality in (3.5) leads to
Therefore, by virtue of the continuity and boundedness of E(t) and ξ(t) on the interval [0, t_{*}], we infer that
for some positive constants K and μ.
Similar to those remarks as in [25], we have the following remark.
Remark. Note that there is a wide class of relaxation functions satisfying (A3). More precisely,
if ξ(t) = e^{αt}, α > 0, then η(t) = α, this gives the exponential decay estimate
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The authors would like to thank very much the anonymous referees for their valuable comments on this work.
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