Elastic materials, when subjected to a suddenly applied loading state held constant thereafter, respond instantaneously with a state of deformation which remains constant. On the other hand, Newtonian viscous fluids respond to a suddenly applied state of uniform shear stress by a steady flow process. However, there exist materials for which any suddenly applied and maintained state of uniform stress produces an instantaneous deformation followed by a flow process which might or might not be limited in magnitude as time grows. Such materials exhibit both instantaneous elasticity effects and creep characteristics. Obviously, such a behavior cannot be described by either an elasticity theory or a viscosity theory only but it combines features of each. The most interesting examples of such materials are polymers, which can display all the intermediate range of properties (glassy, brittle solid or an elastic rubber or a viscous liquid) depending on temperature and the experimentally chosen time scale. Such materials are said to possess memories.
Many scientists, such as Maxwell, Kelvin, Voigt, and Boltzmann, have contributed in modeling these phenomena. Boltzmann, in 1874, supplied the first formulation of a three-dimensional theory of isotropic viscoelasticity. He elaborated the model of a “linear” viscoelastic solid on a basic assumption which states that at any (fixed) point x of the body, the stress at any time t depends on the strain at all the proceeding times. In addition, if the strain at all preceding times is in the same direction, then the effect is to decrease the corresponding stress. The influence of a previous strain on the stress depends on the time elapsed since that strain occurred and is weaker than those strains that occurred long ago. Such properties make the model of solid, elaborated by Boltzmann, a material with (fading) memory. These memory effects are expressed by the dependence on the deformation gradient. Therefore, for these “viscoelastic” materials, the stress at each point and at each instant does not depend only on the present value of the deformation gradient but on the entire temporal prehistory of the motion. In addition, Boltzmann made the assumption that a superposition of the influence of previous strains holds, which means that the stress-strain relation is linear. Mathematically, this is interpreted by the time convolution of a “relaxation” function with the Laplacian of the solution. As a consequence, a subtle damping effect is produced. The types of equations we intend to discuss in this chapter are of the form:
where Ω is a bounded domain with regular boundary, g is a nonincreasing positive function, referred to as the relaxation function which describes the viscoelastic material in consideration, f is an external force, and u(x, t) is the position of a point x “in the reference configuration” at a time t.
In early 1970s, Dafermos [1, 2] discussed a one-dimensional viscoelastic model, where he proved, for smooth monotonically decreasing relaxation functions, various existence and asymptotic stability results. However, no rate of decay has been given. After that, viscoelastic problems have attracted the attention of many researchers and many results of existence and long-time behavior have been established. To the best of our knowledge, the first work that studied the uniform decay of solutions was presented by Dassios and Zafirapoulos . In their work, Dassios and Zafirapoulos presented a viscoelastic problem in ℝ3 and proved a polynomial decay for exponentially decaying kernels. In 1994, Muñoz Rivera  considered, in ℝn and in bounded domains, equations for linear isotropic homogeneous viscoelastic solids, with exponentially decaying memory kernels and showed that, in the absence of body forces, solutions decay exponentially for the bounded-domain case, whereas, for the whole space case, the decay is of a polynomial rate. After that, Cabanillas and Muñoz Rivera  studied problems, where the kernels are of algebraic (but not exponential) decay rates and showed that the decay of solutions is algebraic at a rate which can be determined by the rate of the decay of the relaxation function and the regularity of solutions. This result was later improved by Barreto et al. , where equations related to linear viscoelastic plates were treated. For viscoelastic systems with localized frictional dampings, Cavalcanti et al.  considered the following problem:
where Ω is a bounded domain of ℝn (n ≥ 1) with a smooth boundary ∂Ω, g is a positive nonincreasing function satisfying, for two positive constants, the conditions:
and a(x) ≥ a0 > 0 in a subdomain ω ⊂ Ω, with meas(ω) > 0 and satisfying some geometry restrictions. They established an exponential rate of decay. Berrimi and Messaoudi  improved Cavalcanti’s result by weakening the conditions on both a and g. In particular, the function a can vanish on the whole domain Ω and consequently the geometry condition is no longer needed. This result has been later extended to a situation, where a source is competing with the viscoelastic dissipation, by Berrimi and Messaoudi . Also, Cavalcanti et al.  have studied a quasilinear equation, in a bounded domain, of the form:
with ρ > 0, and a global existence result for γ ≥ 0, as well as an exponential decay for γ > 0, have been established. Messaoudi and Tatar [11,12] discussed the situation when γ = 0 and established polynomial and exponential decay results in the presence, as well as in the absence, of a nonlinear source term. Fabrizio and Polidoro  studied a homogeneous viscoelastic equation in the presence of a linear frictional damping (aut, a > 0) and showed that the exponential decay of the relaxation function g is a necessary condition for the exponential decay of the solution energy of the solution. In other words, the presence of the memory term, with a non-exponentially decaying relaxation function, may prevent the exponential decay even if the frictional damping is linear. He also obtained a similar result for the polynomial decay case.
For more general decaying kernels, Messaoudi [14,15] considered
with b = 0 and b = 1 and for relaxation functions satisfying
where ξ: ℝ+→ℝ+ is a nonincreasing differentiable function. He showed that the rate of the decay of the energy is exactly the rate of decay of g, which is not necessarily of exponential or polynomial decay type. After that, a series of papers using Eq. (1.2) have appeared. See for instance, Han and Wang , Liu [17,18], Park and Park , and Xiaosen and Mingxing .
In this work, we intend to study the following problem:
where Ω is a bounded and regular domain of ℝn, a > 0 is a constant, and g is a positive nonincreasing function satisfying Eq. (1.2). We will establish some general decay results depending on the behavior of g and m.
In this section, we present some material needed in the proof of our result and state a global existence result which can be proved using the well-known Galerkin method. See, for example, [2,3]. In order to prove our main result, we make the following assumptions:
(A1) g: ℝ+ → ℝ+ is a bounded differentiable function such that
where γ(t) is a differentiable function satisfying
(A2) Concerning the nonlinearity in the damping, we assume that
Remark 2.1. Examples of functions satisfying (A1) are
for a and b constants to be chosen properly.
Proposition 2.1. Let be given. Assume that (A1), (A2) hold. Then problem (1.3) has a unique global solution:
Proposition 2.2.  Let E: ℝ+ → ℝ+ be a non-increasing function and φ: ℝ+ → ℝ+ be an increasing C2-function such that
Assume that there exist q ≥ 0 and A > 0 such that
then we have
where c and ω are positive constants independent of the initial energy E(0).
Next, we introduce the “modified energy”:
Remark 2.2. By multiplying Eq. (1.3) by ut and integrating over Ω, using integration by parts and hypotheses (A1), (A2), we get, after some manipulations, as in [3,20],
3. Decay of solutions
In order to state and prove our main result, we set
where ε1 and ε2 are positive constants to be specified later and
Lemma 3.1. For ε1 and ε2 so small, we have
holds for two positive constants α1 and α2.
Proof. It is straightforward to see that
where Cp is the Poincaré constant. In the other hand,
for ε1 and ε2 small enough. Thus, Eq. (3.2) is established.
Lemma 3.2. Assume that m ≥ 2 and assumptions (A1), (A2) hold. Then, the functional Ψ(t) satisfies, along the solution of Eq. (1.3), the estimate:
where C is a “generic” positive constant independent of t.
Proof. By using Eq. (1.3), we easily see that
We now estimate the third term of the right-hand side of Eq. (3.4), using Young’s inequality and (A2). Thus, we get
where cδ is a constant depending on δ. For the fourth term of the right-hand side of Eq. (3.4), we get
We then use Cauchy-Schwarz inequality, Young’s inequality, and the fact that
to obtain, for any η > 0,
By combining Eqs. (3.4)–(3.7), we arrive at
By choosing η = l/(1 − l) and (0), estimate (3.3) is established.
Lemma 3.3. Assume that m ≥ 2 and assumptions (A1), (A3) hold. Then, the functional (t) satisfies, along the solution of Eq. (1.3) and for any δ > 0, the estimate
Proof: By using Eq. (1.3), we easily see that
Similarly to Eq. (3.4), we estimate the right-hand side terms of Eq. (3.9). So for any δ > 0, we have
A combination of Eqs. (3.9)–(3.13), then, yields Eq. (3.8).
Theorem 3.4. Let L2 (Ω) be given. Assume that (A1), (A2) hold. Then, for any t0 > 0, there exist positive constants K and λ such that the solution of Eq. (1.3) satisfies
Proof: We start with the case m ≥ 2. Since g(0) > 0, then there exists t0 > 0 such that
By using Eqs. (2.3), (3.1), (3.3), and (3.8), we obtain
At this point, we choose δ so small that
Whence δ is fixed, the choice of any two positive constants ε1 and ε2 satisfying
We then pick ε1 and ε2 so small that Eqs. (3.2) and (3.16) remain valid and, further,
Therefore, we arrive at
for two constants c, β > 0. We multiply (3.17) by γ(t) and use Eqs. (1.2) and (2.3), to get
This implies that
Again, by using the fact that γ′(t) ≤ 0, letting
and noting that ~, we arrive at
A simple integration of Eq. (3.18) over (t0, t) leads to
We obtain, then, Eq. (3.14) by virtue of equivalence of and .
To establish Eq. (3.15), we re-estimate Eqs. (3.5) and (3.12), for m < 2, as follows
Similarly, we have
By repeating all above steps and using Eqs. (3.19), (3.20) instead of Eqs. (3.5), (3.12), we arrive at
By multiplying Eq. (3.21) by γ(t)q(t), for q > 0 to be specified later, and using (A1), Eq. (2.3), and Young’s inequality, we get
By choosing q = (2 − m)/(2m − 2) (hence, qm/(2 − m) = q + 1] and taking μ small enough, Eq. (3.22) yields
By recalling that γ ′ (t) ≤ 0 and integrating (3.23) over (S, T), S ≥ t0, we get
for some positive constant A. Therefore, Proposition 2.2 gives (3.15). This completes the proof.
Remark 3.1. Estimates (3.14) and (3.15) also hold for t ∈ [0, t0] by virtue of continuity and boundedness of and γ.
Remark 3.2. This result generalizes and improves many results in the literature. In particular, it allows some relaxation functions which satisfy
instead of the usual assumption 1 ≤ ρ < 3/2.
Remark 3.3. Note that the exponential and the polynomial decay estimates, given in early works, are only particular cases of Eq. (3.14). More precisely, we obtain exponential decay for γ(t)≡a and polynomial decay for γ(t) = a(1 + t)− 1, where a > 0 is a constant.
The author would like to express his sincere thanks to King Fahd University of Petroleum and Minerals for its support. This work has been funded by KFUPM under project # IN151002.