Abstract
In this paper, we consider the system of nonlinear viscoelastic equations
with initial and Dirichlet boundary conditions. We prove that, under suitable assumptions on the functions g_{i}, f_{i }(i = 1, 2) and certain initial data in the stable set, the decay rate of the solution energy is exponential. Conversely, for certain initial data in the unstable set, there are solutions with positive initial energy that blow up in finite time.
2000 Mathematics Subject Classifications: 35L05; 35L55; 35L70.
Keywords:
decay; blowup; positive initial energy; viscoelastic wave equations1. Introduction
In this article, we study the following system of viscoelastic equations:
where Ω is a bounded domain in ℝ^{n }with a smooth boundary ∂Ω, and g_{i}(·) : ℝ_{+ }→ ℝ_{+}, f_{i}(·, ·): ℝ^{2 }→ ℝ (i = 1, 2) are given functions to be specified later. Here, u and v denote the transverse displacements of waves. This problem arises in the theory of viscoelastic and describes the interaction of two scalar fields, we can refer to Cavalcanti et al. [1], Messaoudi and Tatar [2], Renardy et al. [3].
To motivate this study, let us recall some results regarding single viscoelastic wave equation. Cavalcanti et al. [4] studied the following equation:
for a : Ω → ℝ^{+}, a function, which may be null on a part of the domain Ω. Under the conditions that a(x) ≥ a_{0 }> 0 on Ω_{1 }⊂ Ω, with Ω_{1 }satisfying some geometry restrictions and
the authors established an exponential rate of decay. This latter result has been improved by Cavalcanti and Oquendo [5] and Berrimi and Messaoudi [6]. In their study, Cavalcanti and Oquendo [5] considered the situation where the internal dissipation acts on a part of Ω and the viscoelastic dissipation acts on the other part. They established both exponential and polynomial decay results under the conditions on g and its derivatives up to the third order, whereas Berrimi and Messaoudi [6] allowed the internal dissipation to be nonlinear. They also showed that the dissipation induced by the integral term is strong enough to stabilize the system and established an exponential decay for the solution energy provided that g satisfies a relation of the form
Cavalcanti et al. [1] also studied, in a bounded domain, the following equation:
ρ > 0, and proved a global existence result for γ ≥ 0 and an exponential decay for γ > 0. This result has been extended by Messaoudi and Tatar [2,7] to the situation where γ = 0 and exponential and polynomial decay results in the absence, as well as in the presence, of a source term have been established. Recently, Messaoudi [8,9] considered
for b = 0 and b = 1 and for a wider class of relaxation functions. He established a more general decay result, for which the usual exponential and polynomial decay results are just special cases.
For the finite time blowup of a solution, the single viscoelastic wave equation of the form
in Ω × (0, ∞) with initial and boundary conditions has extensively been studied. See in this regard, Kafini and Messaoudi [10], Messaoudi [11,12], Song and Zhong [13], Wang [14]. For instance, Messaoudi [11] studied (1.2) for h(u_{t}) = au_{t}^{m2}u_{t }and f(u) = bu^{p2}u and proved a blowup result for solutions with negative initial energy if p > m ≥ 2 and a global result for 2 ≤ p ≤ m. This result has been later improved by Messaoudi [12] to accommodate certain solutions with positive initial energy. Song and Zhong [13] considered (1.2) for h(u_{t}) = Δu_{t }and f(u) = u^{p2}u and proved a blowup result for solutions with positive initial energy using the ideas of the "potential well'' theory introduced by Payne and Sattinger [15].
This study is also motivated by the research of the wellknown KleinGordon system
which arises in the study of quantum field theory [16]. See also Medeiros and Miranda [17], Zhang [18] for some generalizations of this system and references therein. As far as we know, the problem (1.1) with the viscoelastic effect described by the memory terms has not been well studied. Recently, Han and Wang [19] considered the following problem
where Ω is a bounded domain with smooth boundary ∂Ω in ℝ^{n}, n = 1, 2, 3. Under suitable assumptions on the functions g_{i}, f_{i }(i = 1, 2), the initial data and the parameters in the equations, they established several results concerning local existence, global existence, uniqueness, and finite time blowup (the initial energy E(0) < 0) property. This latter blowup result has been improved by Messaoudi and SaidHouari [20], to certain solutions with positive initial energy. Liu [21] studied the following system
where Ω is a bounded domain with smooth boundary ∂Ω in ℝ^{n}, γ_{1}, γ_{2 }≥ 0 are constants and ρ is a real number such that 0 < ρ ≤ 2/(n  2) if n ≥ 3 or ρ > 0 if n = 1, 2. Under suitable assumptions on the functions g(s), h(s), f(u, v), k(u, v), they used the perturbed energy method to show that the dissipations given by the viscoelastic terms are strong enough to ensure exponential or polynomial decay of the solutions energy, depending on the decay rate of the relaxation functions g(s) and h(s). For the problem (1.1) in ℝ^{n}, we mention the work of Kafini and Messaoudi [10].
Motivated by the above research, we consider in this study the coupled system (1.1). We prove that, under suitable assumptions on the functions g_{i}, f_{i }(i = 1, 2) and certain initial data in the stable set, the decay rate of the solution energy is exponential. Conversely, for certain initial data in the unstable set, there are solutions with positive initial energy that blow up in finite time.
This article is organized as follows. In Section 2, we present some assumptions and definitions needed for this study. Section 3 is devoted to the proof of the uniform decay result. In Section 4, we prove the blowup result.
2. Preliminaries
First, let us introduce some notation used throughout this article. We denote by  · _{q }the L^{q}(Ω) norm for 1 ≤ q ≤ ∞ and by ∇ · _{2 }the Dirichlet norm in which is equivalent to the H^{1}(Ω)norm. Moreover, we set
as the usual L^{2}(Ω) inner product.
Concerning the functions f_{1}(u, v) and f_{2}(u, v), we take
where a, b > 0 are constants and p satisfies
One can easily verify that
where
For the relaxation functions g_{i}(t) (i = 1, 2), we assume
(G1) g_{i}(t) : ℝ_{+ }→ ℝ_{+ }belong to C^{1}(ℝ_{+}) and satisfy
and
We next state the local existence and the uniqueness of the solution of problem (1.1), whose proof can be found in Han and Wang [19] (Theorem 2.1) with slight modification, so we will omit its proof. In the proof, the authors adopted the technique of Agre and Rammaha [22] which consists of constructing approximations by the FaedoGalerkin procedure without imposing the usual smallness conditions on the initial data to handle the source terms. Unfortunately, due to the strong nonlinearities on f_{1 }and f_{2}, the techniques used by Han and Wang [19] and Agre and Rammaha [22] allowed them to prove the local existence result only for n ≤ 3. We note that the local existence result in the case of n > 3 is still open. For related results, we also refer the reader to SaidHouari and Messaoudi [23] and Messaoudi and SaidHouari [20]. So throughout this article, we have assumed that n ≤ 3.
Theorem 2.1. Assume that (2.1) and (G1) hold, and that , . Then problem (1.1) has a unique local solution
for some T > 0. If T < ∞, then
Finally, we define
such functionals we could refer to Muñoz Rivera [24,25]. We also define the energy function as follows
where
3. Global existence and energy decay
In this section, we deal with the uniform exponential decay of the energy for system (1.1) by using the perturbed energy method. Before we state and prove our main result, we need the following lemmas.
Lemma 3.1. Assume (2.1) and (G1) hold. Let (u, v) be the solution of the system (1.1), then the energy functional is a decreasing function, that is
Moreover, the following energy inequality holds:
Lemma 3.2. Let (2.1) hold. Then, there exists η > 0 such that for any , we have
Proof. The proof is almost the same that of SaidHouari [26], so we omit it here. □
To prove our result and for the sake of simplicity, we take a = b = 1 and introduce the following:
where η is the optimal constant in (3.3). The following lemma will play an essential role in the proof of our main result, and it is similar to a lemma used first by Vitillaro [27], to study a class of a single wave equation, which introduces a potential well.
Lemma 3.3. Let (2.1) and (G1) hold. Let (u, v) be the solution of the system (1.1). Assume further that E(0) < E_{1 }and
Then
Proof. We first note that, by (2.5), (3.3) and the definition of B, we have
where . It is not hard to verify that g is increasing for 0 < α < α*, decreasing for α > α*, g(α) →  ∞ as α → +∞, and
where α* is given in (3.4). Now we establish (3.6) by contradiction. Suppose (3.6) does not hold, then it follows from the continuity of (u(t), v(t)) that there exists t_{0 }∈ (0, T) such that
By (3.7), we observe that
This is impossible since E(t) ≤ E(0) < E_{1 }for all t ∈ [0, T). Hence (3.6) is established. □
The following integral inequality plays an important role in our proof of the energy decay of the solutions to problem (1.1).
Lemma 3.4. [28]Assume that the function φ : ℝ^{+ }∪ {0} → ℝ^{+ }∪ {0} is a nonincreasing function and that there exists a constant c > 0 such that
for every t ∈ [0, ∞). Then
for every t ≥ c.
Theorem 3.5. Let (2.1) and (G1) hold. If the initial data , satisfy E(0) < E_{1 }and
where the constants α_{*}, E_{1 }are defined in (3.4), then the corresponding solution to (1.1) globally exists, i.e. T = ∞. Moreover, if the initial energy E(0) and k such that
where k = min{k_{1}, k_{2}}, then the energy decay is
for every t ≥ aC^{1}, where C is some positive constant.
Proof. In order to get T = ∞, by (2.2), it suffices to show that
is bounded independently of t. Since E(0) < E_{1 }and
it follows from Lemma 3.3 that
which implies that
where we have used (3.3). Furthermore, by (2.3) and (2.4), we get
from which, the definition of E(t) and E(t) ≤ E(0), we deduce that
for t ∈ [0, T). So it follows from (16) and Lemma 3.1 that
which implies
where C is a positive constant depending only on p.
Next we want to derive the decay rate of energy function for problem (1.1). By multiplying the first equation of system (1.1) by u and the second equation of system (1.1) by v, integrating over Ω × [t_{1}, t_{2}] (0 ≤ t_{1 }≤ t_{2}), using integration by parts and summing up, we have
which implies
For the 11th term on the righthand side of (3.10), one has
Similarly,
Combining (3.10), (3.11) with (3.12), we have
Now we estimate every term of the righthand side of the (3.13). First, by Hölder's inequality and Poincaré's inequality
where λ being the first eigenvalue of the operator  Δ under homogeneous Dirichlet boundary conditions. Then, by (3.9), we see that
where c_{1 }is a constant independent on u and v, from which follows that
Since 0 ≤ J (t) ≤ E (t), from (3.2) we deduce that
Hence, by Poincaré inequality we get
where c_{2 }is a constant independent on u and v. In addition, using Young's inequality for convolution f * g _{q }≤  f _{r }g_{s }with 1/q = 1/r + 1/s  1 and 1 ≤ q, r, s ≤ ∞, noting that if q = 1, then r = 1 and s = 1, we have
and
Hence, combining (3.9), (3.16) with (3.17) we then have
From (3.9), we also have
Combining (3.18) with (3.19), we deduce that
Finally, we also have the following estimate
where c_{3 }is a constant independent on u and v. Combining (3.13)(3.21), we obtain
where C is a constant independent on u.
On the other hand, from (3.3) and (3.9), we have
which implies
Note that E(0) < E_{1}, we see that
Thus, combining (3.22) with (3.23), we have
that is
Denote
We rewrite (3.24)
for every t ∈ [0, ∞).
Since a > 0 from the assumption conditions, by Lemma 3.4, we obtain the following energy decay for problem (1.1) as
for every t ≥ Ca ^{1}. □
4. Blowup of solution
In this section, we deal with the blowup solutions of the system (1.1). Set
From the assumption (G2), we have θ_{i }> 0 (i = 1, 2). Similarly Lemma 3.2, we have
Lemma 4.1. Assume (2.1) holds. Then there exists η_{1 }> 0 such that for any , we have
where the constants θ_{i }(i = 1, 2) are defined in (4.1).
To prove our result and for the sake of simplicity, we take a = b = 1 and introduce the following:
Then we have
Lemma 4.2. Let (G1), G(2) and (2.1) hold. Let (u, v) be the solution of the system (1.1). Assume further that E(0) < E_{2 }and
where the constants θ_{i }(i = 1, 2) are defined in (4.1). Then there exists a constant α_{2 }> α_{* }such that
Proof. We first note that, by (2.5), (4.2) and the definition of B_{1}, we have
where . It is not hard to verify that g is increasing for 0 < α < α_{*}, decreasing for α > α_{*}, g(α) →  ∞ as α → + ∞, and
where α_{* }is given in (4.3). Since E(0) < E_{2}, there exists α_{2 }> α_{* }such that g(α_{2}) = E(0).
Set , then by (4.6) we get g(α_{0}) ≤ E(0) = g (α_{2}), which implies that α_{0 }≥ α_{2}. Now, to establish (4.5), we suppose by contradiction that
for some t_{0 }> 0. By the continuity of we can choose t_{0 }such that
Again, the use of (4.6) leads to
This is impossible since E(t) ≤ E(0) for all t ∈ [0, T). Hence (4.5) is established. □
Theorem 4.3. Assume (G1), (G2) and (2.1) hold. Then any solution of problem (1.1) with initial data satisfying
blows up in finite time, where the constants θ_{i }(i = 1, 2) are defined in (4.1) and α_{*}, E_{2 }are defined in (4.3).
Proof. Assume by contradiction that the solution (u, v) is global. Then, for any T > 0 we consider H(t) : [0, T] → ℝ_{+ }defined by
where β and s_{0 }are positive constants to be determined later. A direct computation yields
and
Multiplying the first equation of system (1.1) by u and the second equation of system (1.1) by v, integrating over Ω, using integration by parts and summing up, we have
which implies
Therefore, we have
where Ψ (t), G(t): [0, T] → ℝ_{+ }are the functions defined by
and
Using the Schwarz inequality, we have
and
Similarly, we have
and
The previous inequalities entail G(t) ≥ 0 for every [0, T]. Using (4.7), we get
where
For the fifth term on the righthand side of (4.9), we have
Similarly,
Combining (4.9), (4.10) with (4.11), we get
Since
and
inserting (4.13) and (4.14) into (4.12), we have
Using (3.2) for s = 0, we have
Since
by Lemma 4.2, there exists a constant α > α_{* }such that
which implies
Thus, we can let β satisfy
which implies that there exists δ > 0 (independent of T) such that
From (4.15) and the definition of H(t), there also exists ρ > 0 (independent of T) such that
By (4.8), (4.16) and (4.17) it follows that
Moreover, we let s_{0 }satisfy that
which means H' (0) > 0. Thus by H" (t) > 0 we see that H(t) and H'(t) is strictly increasing on [0, T].
Setting y(t) = H(t)^{(p+1)/2}, then we have
and
for all t ∈ [0, T], which implies that y(t) reaches 0 in finite time, say as t → T*. Since T* is independent of the initial choice of T, we may assume that T* < T. This tells us that
In turn, this implies that
Indeed, if
then (4.18) immediately follows. On the contrary, if remains bounded on [0, T*), then
so that again (4.18) is satisfied. This implies a contradiction, i.e. T < ∞. □
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
FL and HGAO carried out all studies in this article.
Acknowledgements
The authors are indebted to the referee for giving some important suggestions which improved the presentations of this article. The study was supported in part by the China NSF Grant No. 10871097, the Qing Lan Project of Jiangsu Province, the Foundation for Young Talents in College of Anhui Province Grant No. 2011SQRL115 and the program sponsored for scientific innovation research of college graduate in Jangsu Province Grant No. 181200000649.
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