Global existence and asymptotic behavior of solutions for a system of higher-order Kirchhoff-type equations

This paper deals with the global existence and energy decay of solutions for some coupled system of higher-order Kirchhoff-type equations with nonlinear dissipative and source terms in a bounded domain. We prove the existence of global solutions for this problem by constructing a stable set in H1 0 (Ω)× H m2 0 (Ω) and give the decay estimate of global solutions by applying a lemma of V. Komornik.

When m 1 = m 2 = 1, (1.1)-(1.6)becomes the following initial-boundary value problem for the system of nonlinear wave equations of Kirchhoff-type: ) The equation (1.7)-(1.8)has its origin in the nonlinear vibrations of an elastic string [19].Many authors have investigated the global existence and uniqueness of solutions to the problem related to the system (1.7)-(1.8)through various approaches and assumptive conditions.L. Liu and M. Wang [15] have dealt with the global existence for regular and weak solutions for the problem (1.7)-(1.11)by using Galerkin method.When the initial energy E(0) is non-positive or positive, applying the concavity method [12,13] and the potential well method [3,26,28], they proved the blow-up of solutions in finite time, and give some estimates for the lifespan of solutions.When Φ(s) = s γ , γ ≥ 1, J. Y. Park and J. J. Bae [22] studied the existence and uniform decay of strong solutions of the problem (1.7)- (1.11).In [23,24], they showed the global existence and asymptotic behavior of solutions of the problem (1.7)-(1.11)under some restrictions on the initial energy.S. T. Wu and L. Y. Tsai [30] considered the system (1.7)-(1.11)with Φ( ∇u 2 + ∇v 2 ) = Φ( ∇u 2 ) in (1.7) and Φ( ∇u 2 + ∇v 2 ) = Φ( ∇v 2 ) in (1.8), respectively.They obtain the existence of local and global solutions and give the blowup result for small positive initial energy.When nonlinear dissipative terms in (1.7) and (1.8) become the strong dissipative terms, S. T. Wu [31] discusses the existence, asymptotic behavior and blow-up of solutions of the problem (1.7)-(1.11)under some conditions.Moreover, he gives the decay estimates of the energy function and the estimates for the lifespan of solutions.
For the initial boundary value problem of a single nonlinear higher-order wave equation of Kirchhoff-type data with negative initial energy, the solution blows up at finite time in L γ+2 norms.Later, S. A. Messaoudi and B. Said-Houari [18] improved the results in [14] by modification of the proof and showed the same result when the initial energy has an upper bound.Meanwhile, V. A. Galaktionov and S. I. Pohozaev [6] proved the global existence and nonexistence results of solutions for the Cauchy problem of equation (1.12) without the dissipation (i.e., (1.12) without the term a|u t | q−2 u t ) in the whole space R n .However, their approach can not be applied to the problem (1.12)- (1.14).Motivated by the above researches, in this paper, we prove the global existence for the problem (1.1)-(1.6)by constructing a stable set in H m 1 0 (Ω) × H m 2 0 (Ω) and give the energy decay of global solutions by applying a lemma of V. Komornik [11].
We adopt the usual notations and convention.Let H m (Ω) denote the Sobolev space with the usual scalar products and norm.Meanwhile, For simplicity of notations, hereafter we denote by • r the Lebesgue space L r (Ω) norm and [2,5,8]).Moreover, C i (i = 0, 1, 2, 3, . . . ) denotes various positive constants which depend on the known constants and may be different at each appearance.
This paper is organized as follows: in the next section, we give some preliminaries.In Section 3, we prove the existence of global solutions for problem (1.1)-(1.6).The Section 4 is devoted to the study of the energy decay of global solutions.

Preliminaries
To state and prove our main results, we make the following assumptions: Concerning the functions f 1 (u, v) and f 2 (u, v), we assume that where b 1 , b 2 > 0 and p > 1 are constants.
It is easy to see that where Moreover, a quick computation will show that there exist two positive constants C 0 and C 1 such that the following inequality holds (see [17]) Now, we define the following functionals: ) Then we can define the stable set W of the problem (1.1)-(1.6)as follows We denote the total energy related to the equations (1.1) and (1.2) by is the initial total energy.We state some known lemmas which will be needed later.
Lemma 2.1.Let r be a number with 2 ≤ r < +∞ if n ≤ 2m and 2 ≤ r ≤ 2n n−2m if n > 2m.Then there is a constant C depending on Ω and r such that Lemma 2.2 (Young's inequality).Let X, Y and ε be positive constants and ς, σ (2.9) Proof.Multiplying equation (1.1) by u t and (1.2) by v t , and integrating over Ω × [0, t], then, adding them together, and integrating by parts, we get Being the primitive of an integrable function, E(t) is absolutely continuous and equality (2.9) is satisfied.
The local existence and uniqueness of solutions for the problem (1.1)-(1.6)can be obtained by a similar way as done in [1,7,16,20,21,27,32].The result reads as follows.

Global existence of solutions
The following lemmas play an important role in the proof of global existence of solutions.
The main result in this section reads as follows.
Theorem 3.3 (Global solutions).Suppose that (3.3), (A1) and (A2) hold, and Existence and asymptotic behavior for a system of Kirchhoff equations 7 Therefore, we get The above inequality and the standard continuation principle [9,29] lead to the global existence of the solution, that is, T = +∞.Hence, the solution [u, v] is a global solution of the problem (1.1)-(1.6).

Energy decay of global solution
In order to study the decay estimate of global solutions for the problem (1.1)-(1.6),we need the following lemma.Lemma 4.1 ([6]).Let Y(t) : R + → R + be a nonincreasing function and assume that there are two constants η ≥ 1 and M > 0 such that where C and ω are positive constants independent of Y(0).
The following result is concerned with the energy decay estimate of global solutions for the problem (1.1)- (1.6).The theorem reads as follows.
where K > 0 is a constant depending on initial energy E(0).

( 4 . 3 )
It suffices to show that u t 2 + v t 2 + D m 1 u 2 + D m 2 v 2 isbounded independently of t.Under the hypotheses in Theorem 3.3, we get from Lemma 3.2 that [u, v] ∈ W on [0, T).