GLOBAL SOLVABILITY AND GENERAL DECAY OF A TRANSMISSION PROBLEM FOR KIRCHHOFF-TYPE WAVE EQUATIONS WITH NONLINEAR DAMPING AND DELAY TERM

. A transmission problem for Kirchhoff-type wave equations with nonlinear damping and delay term in the internal feedback is considered under a memory condition on one part of the boundary. By virtue of multiplier method, Faedo-Galerkin approximation and energy perturbation technique, we establish the appropriate conditions to guarantee the existence of global solution, and derive a general decay estimate of the energy, which includes exponential, algebraic and logarithmic decay etc.

It is well known that delay effect, which arises in many practical problems, may be the source of instability. Hence, the control of PDEs with time delay has become an active area of research in recent years. For example, it was proved in [17,18,36,37,44] that an arbitrarily small delay may destabilize a system which is uniformly asymptotically stable, unless additional conditions or control terms were used. For the influence of linear delay term on the stability, we see [1,22,23,27,31]. A boundary stabilization problem for the wave equation with interior delay was studied in [1]. The authors derived an exponential stability result under some Lions geometric condition. Kirane and Said-Houari [23] considered the viscoelastic wave equation with a delay where µ 1 and µ 2 are positive constants. Under the hypothesis of 0 ≤ µ 1 ≤ µ 2 , they established a general decay estimate of the energy. Later, Liu [31] improved this result by considering the equation with a time-varying delay whose coefficient µ 2 is not necessarily positive, and Feng [22] studied the plate equation. Furthermore, we refer to [27] for the decay of a weak viscoelastic equation with time-varying delay.
However, there are few results on the effect of nonlinear delay, which can be seen in [8,9,20,25]. Benaissa and Louhibi [9] considered the following wave equation with nonlinear delay where β : R → R is a nondecreasing continuous function satisfying some appropriate nonlinear conditions. Under Dirichlet boundary condition, they derived the global existence and asymptotic behavior of the solution. In [8], Benaissa et al. obtained the uniform decay estimate of the Dirichlet initial boundary value problem for the following viscoelastic equation with nonlinear delay in the internal feedback Recently, Li and Chai [25] studied the Euler-Bernoulli plate equation with localized nonlinear internal feedback They established the energy decay estimate by Riemannian geometric method. Djilali [20] considered the nonlinear Timoshenko beam system with nonlinear delay in one-dimensional space and deduced the global existence and uniform decay rate of the energy.

ZHIQING LIU AND ZHONG BO FANG
For the transmission problem, [2,7,12,19,26,29,[33][34][35] studied the existence, regularity, controllability and decay estimate of solution for the transmission problems with Laplacian operators. For example, Marzocchi [33] proved that the solution for a semilinear transmission problem in one-dimensional space between elastic and thermoelastic materials decays exponentially. This result was extended to N -dimensional space by Marzocchi and Naso [34]. Most recently, Liu et.al [29] established the general decay of the solution for a transmission problem in memorytype thermoelasticity with second sound. Moreover, Bastos and Raposo [7] and Cavalcanti et.al [12] investigated the exponential stability of a transmission problem with frictional damping and a transmission problem of viscoelastic waves with hereditary memory, respectively.
Recently, there are many new results for transmission problems with operators of Kirchhoff-type, see [6,10,24,35,39,40,43]. Bae [6] investigated the transmission problem for wave equations given by . Under a memory condition on a part of the boundary, he studied the global existence of the solution and showed that the energy of the problem have the same decay rate with the relaxation function, which decays exponentially or polynomially. Later, Park [39,40] considered the uniform decay rate of the transmission problem of the Kirchhoff-type wave equations.
On the other hand, for the transmission problem with delay, Benseghir [10] studied a linear transmission problem with a delay term in one-dimensional space where L = (0, L 1 ) ∪ (L 2 , L 3 ). Under the assumption that the effect of delay is weaker than damping (µ 2 < µ 1 ), he showed the exponential stability of the solution by introducing a suitable Lyaponov functional. Li et al. [24] studied the following linear transmission system with long-time memory and delay term By assuming µ 2 ≤ µ 1 , they proved the well-posedness result by using semigroup theory and Hille-Yosida theorem. Furthermore, they established a general decay result, which the exponential and polynomial decay are only special cases. Moreover, we refer to [43] for the similar transmission problem with short-time memory.
In view of the works mentioned above, it is clear that research on the global well-posedness and general uniform energy decay for Kirchhoff-type transmission problem (1.1)-(1.8) has not been started yet. The main difficulties lie in finding the competitive relationship between Kirchhoff-type operators, nonlinear delay, nonlinear damping, and boundary memory term. Motivated by these observations, we investigate the global existence of solution to problem (1.1)-(1.8) and its uniform decay by using multiplier method, Faedo-Galerkin approximation and energy perturbation technique, which includes exponential, algebraic and logarithmic decay etc.
The remainder of this paper is organized as follows. In Sect.2, we present some preliminaries and state the main results. In Sect.3, we prove a global solvability of problem (1.1)-(1.8). A general decay estimate of energy is derived in Sect.4.

Preliminaries and main results.
In this section, we introduce some materials needed in the proof of our results and state the main results.
Throughout this paper, positive constant C is not necessarily the same at each occurrence and we define For a Banach space X, ∥ · ∥ X represents the norm of X. It's convenient to denote ∥ · ∥ L 2 (Ωi) and ∥ · ∥ L 2 (Γj ) by ∥ · ∥ Ωi and ∥ · ∥ Γj , respectively. Moreover, we give some simple notations.
Differentiating (1.3), we arrive at the following Volterra equation By using of the Volterra¡¯s inverse operator, we get where the resolvent kernel satisfies k(t) , then the aforementioned equality can be written as The identity (2.3) implies (1.3).
The following assumptions are made to state the corresponding results. We begin with some assumptions on nonlinear functions β and ϕ.

4)
and and Remark 1. Noting that ϕ is an odd and increasing function, we find The hypothesis (2.6) is only used to prove global existence of the solution to problem (1.1)-(1.8). For the energy estimate, we may remove this restriction of linear growth order on function ϕ.
For the resolvent kernel k, as in [39], we assume that: and there exists a nonincreasing continuous function ξ : Next, we present the definition of weak solution to our problem.
As for the global solvability for problem (1.1)-(1.8) in time, we get the following result.
In order to state the asymptotic behavior of the energy, we define the energy functional as where ζ is a positive constant such that Then, we have the following general decay for (1.1)-(1.8).
Theorem 2. Let (u, v) be the solution of (1.1)- (1.8). Assuming |µ 2 | < µ1 2φ1 and (H1)-(H3) hold. Then for t 0 > 0 large enough, there exist constants C 0 > 0 and > 0 such that Remark 3. The exponential decay and polynomial decay in previous literatures are special cases of the result in Theorem 2. In fact, if we take , then by Theorem 2, the energy may decay exponentially, polynomially, and logarithmically, respectively.
3. Global solvability. In this section, by using Feado-Galerkin approximation technique and multiplier method, we prove Theorem 1.
The proof of Theorem 1: We divide the proof into four steps.
We define the approximations and According to the standard theory of ordinary differential equations, the finite dimensional problem (3.1)-(3.4) has a unique solution (b jn (t)) j=1,...,n defined on [0, T n ), T n > 0. The extension of these solutions to the whole interval [0, T ], for all T > 0, is a consequence of the first estimate which we are going to prove below.
Step 2. Energy estimates. Estimate I: Multiplying (3.1) by b ′ jn (t) and summing on j, then using (2.1) we have d dt where ζ is a positive constant such that Combining (3.5) and (3.6), we can derive where Let's denote Φ * to be the conjugate function of the convex function Φ, i. e., Φ * = sup t∈R + (st − Φ(t)). Then Φ * is the Legendre transform of Φ which is given by (see Arnold [4,) and satisfies the inequality Taking the definition of Φ into account, we get Noting that ϕ is an odd function, from (H2), (3.9), and (3.10), we obtain Moreover, it follows from Young's inequality that Substituting (3.11) and (3.12) into (3.7) to obtain Integrating (3.13) over (0, t), 0 < t ≤ T , and then using Gronwall's inequality and (3.2)-(3.4), we obtain the first estimate where L 1 > 0 is a constant independent of n.
Estimate II: First of all, we are going to estimate the initial data u Noting that β(u 1n ) and ϕ(f 0n (−τ )) are bounded in L 2 (Ω 1 ) by (H1), (H2) and (3. Now, differentiating (3.1) with respect to t, multiplying it by b ′′ jn (t), and summing on j, we have d dt In addition, we have By virtue of Young's inequality, we obtain and (3.26) By combining (3.14), (3.17) and (3.26) we have Integrating the inequality above over (0, t), 0 < t ≤ T sufficiently small, then using nonlinear Gronwall's inequality [21,Thoerem 21,pp.11], we obtain the second estimate 27) where L 2 > 0 is a constant independent of n.
Step 3. Pass to the limit. It follows from the first prior estimate (3.14) and second prior estimate (3.27) that for all T ≥ 0. Noting (H1) and (H2), from (3.28), we also obtain the following estimates (3.29) where C is a positive constant independent of n and t. Therefore, (3.28) and (3.29) permit us to obtain subsequences of {u (n) }, {v (n) } (we still denote the subsequences by {u (n) }, {v (n) } for convenience) such that
With the four steps above, we get the global well-posedness of solution for the problem (1.1)-(1.8).
4. General decay estimate. In this section, by virtue of the energy perturbation technique, we give the proof of our main result Theorem 2 in detail.
Firstly, we introduce the following lemmas which play a key observation in the proof.
Next, we define the functional Then we have the following lemma. where C(τ ) is a positive constant only depending on τ .
Proof. We use the method introduced by [3] to prove this lemma. Taking the derivative of Λ 2 (t) directly, we have where M i (i = 1, 2, 3) are positive constants which will be determined later.