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Arbitrary decays for a viscoelastic equation

Shun-Tang Wu

Author affiliations

General Education Center National Taipei University of Technology Taipei 106, Taiwan

Citation and License

Boundary Value Problems 2011, 2011:28  doi:10.1186/1687-2770-2011-28


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/28


Received:16 February 2011
Accepted:6 October 2011
Published:6 October 2011

© 2011 Wu; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we consider the nonlinear viscoelastic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M1">View MathML</a>, in a bounded domain with initial conditions and Dirichlet boundary conditions. We prove an arbitrary decay result for a class of kernel function g without setting the function g itself to be of exponential (polynomial) type, which is a necessary condition for the exponential (polynomial) decay of the solution energy for the viscoelastic problem. The key ingredient in the proof is based on the idea of Pata (Q Appl Math 64:499-513, 2006) and the work of Tatar (J Math Phys 52:013502, 2010), with necessary modification imposed by our problem.

Mathematical Subject Classification (2010): 35B35, 35B40, 35B60

Keywords:
Viscoelastic equation; Kernel function; Exponential decay; Polynomial decay

1 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 integro-differential 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 blow-up have been obtained, see [1-28] and the reference therein.

In [3], Cavalcanti et al. studied the following problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M2">View MathML</a>

(1.1)

where Ω ⊂ RN, N ≥ 1, is a bounded domain with a smooth boundary ∂Ω, γ ≥ 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M3">View MathML</a> if N ≥ 3 or ρ > 0 if N = 1, 2, and the function g: R+ R+ is a nonincreasing function. This type of equations usually arise in the theory of viscoelasticity when the material density varies according to the velocity. In that paper, they proved a global existence result of weak solutions for γ ≥ 0 and a uniform decay result for γ > 0. Precisely, they showed that the solutions goes to zero in an exponential rate for γ > 0 and g is a positive bounded C1-function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M4">View MathML</a>

(1.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M5">View MathML</a>

(1.3)

for all t ≥ 0 and some positive constants ξ1 and ξ2. 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 [1,2,4-6,13,22,23,29-31]. Lately, these conditions have been weakened by some researchers. For instance, instead of (1.3) Furati and Tatar [8] required the functions eαt g(t) and eαtg'(t) to have sufficiently small L1-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 eαt g(t) ∈ L1(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) ∈ L1(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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M6">View MathML</a>

(1.4)

with initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M7">View MathML</a>

(1.5)

and boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M8">View MathML</a>

(1.6)

where Ω ⊂ RN, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M9">View MathML</a> for some nonnegative function ξ(t). Therefore, our result allows a larger class of relaxation functions and improves some earlier results concerning the exponential decay or polynomial decay.

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 Lp(Ω) and Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M10">View MathML</a> with their usual inner products and norms.

Lemma 2.1. (Sobolev-Poincaré inequality) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M11">View MathML</a>, the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M12">View MathML</a>

holds with the optimal positive constant cs, where || · ||p denotes the norm of Lp(Ω).

Assume that ρ satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M13">View MathML</a>

(2.1)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M14">View MathML</a>

(2.2)

(A2) g'(t) ≤ 0 for almost all t > 0.

(A3) There exists a positive nondecreasing function ξ(t): [0, ∞) → (0, ∞) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M15">View MathML</a> is a decreasing function and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M16">View MathML</a>

(2.3)

Now, we state, without a proof, the existence result of the problem (1.4)-(1.6) which can be established by Faedo-Galerkin methods, we refer the reader to [3,5].

Theorem 2.2. Suppose that (2.1) and (A1) hold, and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M17">View MathML</a>. Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M3">View MathML</a>, if N ≥ 3, p > 0, if N = 1, 2. Then there exists at least one global solution u of (1.4)-(1.6) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M18">View MathML</a>

Next, we introduce the modified energy functional for problem (1.4)-(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M19">View MathML</a>

(2.4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M20">View MathML</a>

(2.5)

Lemma 2.3. Let u be the solution of (1.4)-(1.6), then the modified energy E(t) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M21">View MathML</a>

(2.6)

Proof. Multiplying Eq. (1.4) by ut 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M22">View MathML</a>

(2.7)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M23">View MathML</a>

(3.1)

where λi are positive constants, i = 1, 2, 3 to be specified later and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M24">View MathML</a>

(3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M25">View MathML</a>

(3.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M26">View MathML</a>

(3.4)

here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M27">View MathML</a>

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) + Φ3(t) are equivalent.

Lemma 3.1. There exists two positive constants β1 and β2 such that the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M28">View MathML</a>

(3.5)

holds for all t ≥ 0 and λi small, i = 1, 2.

Proof. By Hölder inequality Young's inequality Lemma 2.1, (2.7) and (2.2), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M29">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M30">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M31">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M32">View MathML</a>. Therefore, from above estimates, the definition of E(t) by (2.4) and (2.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M33">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M34">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M36">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M37">View MathML</a>. Hence, selecting λi , i = 1, 2 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M38">View MathML</a>

and again from the definition of E(t), there exist two positive constants β1 and β2 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M39">View MathML</a>

To obtain a better estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M40">View MathML</a>, we need the following Lemma which repeats Lemma 2 in [25].

Lemma 3.2. For t ≥ 0, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M41">View MathML</a>

(3.6)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M42">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M43">View MathML</a>

The flatness set and the flatness rate of g are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M44">View MathML</a>

(3.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M45">View MathML</a>

(3.8)

Before proceeding, we note that there exists t0 > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M46">View MathML</a>

(3.9)

since g is nonnegative and continuous.

Theorem 3.3. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M17">View MathML</a>be given. Suppose that (A1)-(A3), (2, 1) and the hypothesis on p hold. Assume further that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M49">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M50">View MathML</a>

Then the solution energy of (1.4)-(1.6) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M51">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M52">View MathML</a>

which together with the identity (3.6) and (2.2) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M53">View MathML</a>

(3.10)

Next, we would like to estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M54">View MathML</a>. Taking a derivative of Φ2 in (3.3) and using Eq. (1.4) to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M55">View MathML</a>

(3.11)

We now estimate the first two terms on the right-hand side of (3.11) as in [25].

Indeed, for all measure set A and F such that A = R+ - F, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M56">View MathML</a>

(3.12)

To simplify notations, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M57">View MathML</a>

Using Hölder inequality Young's inequality and (2.2), we see that, for δ1 > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M58">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M59">View MathML</a>

Thus, from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M60">View MathML</a> by (3.8), (3.12) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M61">View MathML</a>

(3.13)

The second term on the right-hand side of (3.11) can be estimated as follows (see [25]), for δ2 > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M62">View MathML</a>

(3.14)

Using Hölder inequality Young's inequality and (A2) to deal with the fifth term, for δ3 > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M63">View MathML</a>

(3.15)

Exploiting Hölder inequality Young's inequality Lemma 2.1 and (A2) to estimate the sixth term, for δ4 > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M64">View MathML</a>

(3.16)

For the last term, thanks to Hölder inequality Young's inequality Lemma 2.1, (2.7), (2.2) and (3.8), we have, for δ5 > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M65">View MathML</a>

(3.17)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M66">View MathML</a>. Thus, gathering these estimates (3.13)-(3.17) and using (3.9), we obtain, for t t0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M67">View MathML</a>

(3.18)

Further, taking a derivative of Φ3(t), using the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M15">View MathML</a> is a decreasing function and the definition of Φ3(t) by (3.4), we derive that (see [25])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M68">View MathML</a>

(3.19)

Hence, we conclude from (2.6), (3.10), (3.18) and (3.19) that for any t t0 > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M69">View MathML</a>

(3.20)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M70">View MathML</a>, we consider the sets (see [24,25])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M71">View MathML</a>

and observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M72">View MathML</a>

where Fg is given in (3.7) and Ng is the null set where g' is not defined. In addition, denoting Fn = R+ - An, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M73">View MathML</a>

because An are increasingly nested. Thus, choosing A = An, F = Fn and λ1 = (g* - ε) λ2 for some ε > 0 in (3.20), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M74">View MathML</a>

(3.21)

At this point, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M75">View MathML</a> and select λ2 so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M76">View MathML</a>

then (3.21) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M77">View MathML</a>

For ε, δ2 small enough and large value of n and t0, we see that if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M78">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M79">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M80">View MathML</a>

(3.22)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M81">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M82">View MathML</a>

Note that α > 0 and 0 < δ < 1 due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M49">View MathML</a>. Furthermore, we require λ2 and λ3 satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M83">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M84">View MathML</a>

this is possible because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M48">View MathML</a>. Then, letting δ1 be small enough and using (3.22), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M85">View MathML</a>

Hence, from the definition of E(t) by (2.4), we have, for all t t0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M86">View MathML</a>

for some positive constant c4. As η(t) is decreasing, we have η(t) ≤ c4 after some t* t0. Hence, with the help of the right hand side inequality in (3.5), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M87">View MathML</a>

(3.23)

for some positive constant c5 > 0. An integration of (3.23) over (t*, t) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M88">View MathML</a>

Then using the left hand side inequality in (3.5) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M89">View MathML</a>

Therefore, by virtue of the continuity and boundedness of E(t) and ξ(t) on the interval [0, t*], we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M90">View MathML</a>

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) = eαt, α > 0, then η(t) = α, this gives the exponential decay estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M91">View MathML</a>, for some positive constants c1 and c2. Similarly, if ξ(t) = (1 + t)α , α > 0, then we obtain the polynomial decay estimate E (t) ≤ c3 (1 + t)-μ, for some positive constants c3 and μ.

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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