Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions

This paper is devoted to the study of a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Based on the Faedo-Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we first establish two local existence theorems of weak solutions. By the construction of a suitable Lyapunov functional, we next prove a blow up result and a decay result of global solutions.

Problems of this type with many results concerning local existence, global existence, decay, and blow-up of solutions for a system of wave equations have been extensively studied by many authors. For example, we refer to [1]- [6], [8,9], [11]- [30], and the references given therein.
In [11], the general decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping were considered. Here, two blow-up results with nonpositive initial energy as well as positive initial energy by exploiting the convexity technique were established and a decay result of global solutions by the perturbed energy method under a weaker assumption on the relaxation functions were proved.
In [17], Messaoudi established a blow up result for solutions with negative initial energy and a global existence result for arbitrary initial data of a nonlinear viscoelastic wave equation associated with initial and Dirichlet boundary conditions.
It is also well known that the single viscoelastic wave equation of the form with initial and boundary conditions, where Ω ⊂ R n is bounded domains with a smooth boundary ∂Ω, has been extensively studied and many results concerning existence, nonexistence, exponential decay and blow up in finite time have been proved. For example, we refer to the series works of Thomas C Sideris, Zhen Lei and references therein [8,9], [25]- [27] about classical solutions to the equations of elasticity and viscoelasticity. In [26], Sideris considered the equations of motion for the displacement of an isotropic, homogeneous, hyperelastic material of the form of a quasilinear hyperbolic system in three space dimensions and the author proved that for certain classes of materials, small initial disturbances gave rise to global smooth solutions. These special materials were distinguished by a null condition imposed on the quadratic portion of the nonlinearity.
In [27], the existence of global classical solutions to the Cauchy problem in nonlinear elastodynamics was studied. The unbounded elastic medium was assumed to be homogeneous, isotropic, and hyperelastic. As in the theory of 3D nonlinear wave equations in three space dimensions, global existence hinges on two basic assumptions. First, the initial deformation must be a small placement from equilibrium, in this case a prestressed homogeneous dilation of the reference configuration, and equally important, the nonlinear must obey a type of nonresonance or null condition. The omission of either of these assumptions can lead to the breakdown of solutions.
In [8], the long-time behavior of elastic waves for isotropic incompressible materials was studied in 2-D. The equations of incompressible elastodynamics displayed a linear degeneracy in the isotropic case; i.e., the equation inherently satisfied a null condition. The authors proved that the 2-D incompressible isotropic nonlinear elastic system was almost globally well-posed for small initial data. More precisely, the authors proved that for initial data of the form U 0 , there exists a unique solution for a time interval [0, exp(T (U 0 )/ε)], where T (U 0 ) depends only on a certain weighted Sobolev norm of the U 0 .
M. M. Cavalcanti et al. [4] studied the existence of global solutions and the asymptotic behavior of the energy related to a degenerate system of wave equations with boundary conditions of memory type. By the construction of a suitable Lyapunov functional, the authors proved that the energy decays exponentially. The same method was also used in [24] to study the asymptotic behavior of the solutions to a coupled system of wave equations having integral convolutions as memory terms. The author showed that the solution of that system decays uniformly in time, with rates depending on the rate of decay of the kernel of the convolutions.
In [16,28], the existence, regularity, blow up and exponential decay estimates of solutions for nonlinear wave equations associated with two-point boundary conditions have been established. In which, the main tools to obtain existence results are the Galerkin method associated to a priori estimates, weak convergence, compactness techniques. On the other hand, a suitable Lyapunov functional was constructed to obtain the blow up and exponential decay results.
The above mentioned works lead to the study of the existence, blow-up and exponential decay estimates for a system of nonlinear wave equations (1)-(3). It consists of four sections. Section 2 is devoted to the presentation of the existence results based on Faedo-Galerkin method and standard arguments of density corresponding to the regularity of initial conditions. In this section, problem (1)-(3) is dealt with two cases of (r 1 , G) : (i)r 1 = 2 and G ∈ C 1 (R) or (ii)r 1 > 2 and G ∈ C 2 (R) and r 1 , r 2 ≥ 2. In the casesr 1 = 2 and G ∈ C 1 (R) orr 1 > 2 and G ∈ C 2 (R) such that G (z) ≤ ζ max < µ 1 * ∀z ∈ R, with µ 1 (x, t) ≥ µ 1 * > 0 for all (x, t) ∈ [0, 1] × R + , the solution obtained here is unique. Because of the difficulties arising in passing to the limit in the nonlinear terms imposed the value on boundary, we need to use the Sobolev imbedding theorem suitably combined with the technical estimates; thus, the problem here is considred in the cases of (r 1 , G) and it is still an open problem in higher-dimensional case.
In Sections 3, 4, problem (1)-(3) is considered with r 1 = r 2 =r 1 = 2 and µ 2 (x, t) =μ 2 (x). Under some suitable conditions, by applying techniques as in [28] with some necessary modifications, we prove that the solution of (1)-(3) blows up in finite time. We also prove that the solution (u(t), v(t)) will exponential decay if the initial energy is positive and small.
2. The existence and uniqueness of a weak solution. The notations we use in this paper is standard and can be found in Lions' book [10], with · for the norm in L 2 and · H 1 for the norm in H 1 . Using the norm v H 1 = ( v 2 + v x 2 ) 1/2 , we have the following lemma, it is a known property.
We now state the first theorem about the existence of a "strong solution". Theorem 2.6. Suppose that (H 1 ) − (H 7 ) hold and the initial data satisfy the compatibility conditions If either of the following cases is valid (i)r 1 = 2 and G ∈ C 1 (R) , or (ii)r 1 > 2 and G ∈ C 2 (R) , then there exists a local weak solution (u, v) of problem (1)-(3) such that for T * > 0 small enough.
Furthermore, if either of the following cases is valid or (ii)r 1 > 2 and G ∈ C 2 (R) , there exists a constant ζ max > 0 with ζmax µ1 * < 1 such that with µ * 1 as in (H 2 , (i)), then the solution is unique.
Remark 2.7. The regularity obtained by (19) shows that problem (1)-(3) has a strong solution With less regular initial data, we have the second theorem about the existence of a weak solution.
hold. If either of the cases given in (18) is valid, then there exists a local weak solution for T * > 0 small enough. Furthermore, if either of the cases given in (20), (21) is valid, then the solution is unique.
Proof of Theorem 2.2. The proof consists of four steps. Step with the coefficient functions (c mj , d mj ) defined via the following system (25) By the assumptions of Theorem 2.2, it is obviously that system (25) has a solution (u m (t), v m (t)) on an interval [0, Step 2. The first estimate. Multiplying the j th equation of (25) by (c mj (t), d mj (t)) and summing with respect to j, and afterwards integrating with respect to the time variable from 0 to t, we get after some rearrangements By (25) 3 and (27), we obtain We need the following lemma.
Lemma 2.9. Let r > 1 and δ > 0. Then The proof is straightforward and we omit the details. We now to estimate the terms of (26) as belows.
Using the inequalities (29) and with we obtain where we note that C T (δ) always indicates a bound depending on T and δ.
where M T is a constant depending only on T as above. (25) 1 , multiplying the result by c mj (0) and using the compatibility (17), we have

T (s) =D
(2) T and C T are the constants depending on T. By solving a nonlinear Volterra integral inequality (74), there exists a constant T * * > 0 depending on T * (independent of m) such that whereM T is a constant depending only on T.
In what follows, we will denote T * for T * * or T * .
Proof of Theorem 2.3. For the existence of a weak solution, we use standard arguments of density. The proof is similar to the arguments in [20], so we omit the details. The uniqueness of a weak solution is obtained by using the well-known regularization procedure due to Lions, see for example Ngoc et al. [23].
By solving nonlinear Volterra integral inequalities as in (47), it ensures that a global estimate will be obtained. However, this global estimate is not enough to pass to the limit in all nonlinear terms, specialy passing to the limit the term |u t (0, t)|r 1−2 u t (0, t) imposed the value at boundary point x = 0, only if r 1 = r 2 = r 1 = 2. Otherwise, we need the second estimate, by solving nonlinear Volterra integral inequalities as in (71), this leads immediately to the global existence of solutions for problem (1)-(3) whenr 1 ≥ 3.
Because of the difficulties arising in passing to the limit in the nonlinear terms imposed the value on boundary, the Sobolev imbedding theorem can not be chosen to apply appropriately; thus, it is still an open problem in higher-dimensional case.
Then we obtain the theorem. Step 1. Constructing Lyapunov functional. We denote by E(t) the energy associated to the solution (u, v), defined by and we put H(t) = −E(t).
for any (u, v) ∈ V × H 1 0 . Proof of Lemma 3.2 is straightforward, so we omit the details.
In order to obtain the decay result, we construct the functional where δ > 0; E(t) and Φ(t) are defined as in (122), (130), respectively.