Existence, Blow-Up and Exponential Decay Estimates for the Nonlinear Kirchho ﬀ -Carrier Wave Equation in an Annular with Nonhomogeneous Dirichlet Conditions

. This paper is devoted to the study of a nonlinear Kirchho ﬀ -Carrier wave equation in an annular associated with nonhomogeneous Dirichlet conditions. At ﬁrst, by applying the Faedo-Galerkin, we prove existence and uniqueness of the solution of the problem considered. Next, by constructing Lyapunov functional, we prove a blow-up result for solutions with a negative initial energy and establish a su ﬃ cient condition to obtain the exponential decay of weak solutions.

It is known that Kirchhoff [6] first investigated the following nonlinear vibration of an elastic string where u = u(x, t) is the lateral displacement at the space coordinate x and the time t, ρ is the mass density, h is the cross-section area, L is the length, E is the Young modulus, P 0 is the initial axial tension.
In [3], Carrier established the equation which models vibrations of an elastic string when changes in tension are not small L 0 u 2 (y, t)dy u xx = 0, (1.5) where u(x, t) is the x−derivative of the deformation, T 0 is the tension in the rest position, E is the Young modulus, A is the cross -section of a string, L is the length of a string and ρ is the density of a material. Clearly, if properties of a material depends on x and t, there is a hyperbolic equation of the type (Larkin [7]) The Kirchhoff -Carrier equations of the form Eq. (1.1) received much attention. We refer the reader to, e.g., Cavalcanti et al. [1], [2], Ebihara, Medeiros and Miranda [4], Miranda et al. [15], Lasiecka and Ong [8], Hosoya, Yamada [5], Larkin [7], Long et al. [10]- [12], Medeiros [14], Menzala [16], Messaoudi [17], Ngoc et al. [18]- [22], Park et al. [23], [24], Rabello et al. [25], Santos et al. [26], Truong et al. [28], for many interesting results and further references. In these works, the results concerning local existence, global existence, asymptotic expansion, asymptotic behavior, decay and blow-up of solutions have been examined.
Recently, Gongwei Liu [13] studied the damped wave equation of Kirchhoff type with initial and Dirichlet boundary condition in Ω × (0, ∞), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ Ω, u(x, t) = 0 on ∂Ω × (0, ∞), where Ω is a bounded domain with smooth boundary ∂Ω, is a source term with exponential growth at the infinity to be specified later. Here M(s) is a positive C 1 function on [0, ∞) and M(s) ≥ 1, |M (s)| ≤ s α for all s > 1, α ≥ 0 and for suitably chosen initial data, (1.7) possesses a global weak solution which decays exponentially. On the other hand, if conditions of M, and initial data are suitable, the solution u of (1.7) blows-up at a finite T * .
In [15] Miranda and Jutuca dealt with the existence and uniqueness of solutions and exponential decay of solutions of an initial-homogeneous boundary value problem for the Kirchhoff equation.
In [1], [2], Cavalcanti also studied the existence and uniform decay of solutions of the Kirchhoff-Carrier equation.
In [28], the global existence and regularity of weak solutions for the linear wave equation with the initial conditions as in (1.3) and two-point boundary conditions. The exponential decay of solutions was given there by using Lyapunov method.
Motivated by the above work, we intend to study the existence and uniqueness, the blow-up and exponential decay of solutions for problem (1.1-1.3) under suitable conditions on f, µ and initial data. Our paper is organized as follows.
First, we present preliminaries in Section 2, with the notations, definitions, list of appropriate spaces and required lemmas. We prove the existence and uniqueness a weak solution in Section 3 by using Faedo-Galerkin method, the linearization method and the weak compact method. Next, in Section 4, Prob.
If some auxiliary conditions are satisfied, we imply that the weak solution u of Prob. (1.1)-(1.3) blows-up at finite time.
Finally, in Section 5 with the case µ = µ u x (t) 2 f (z)dz > 0 and the initial energy and F(t) 0 are small enough, we verify that the energy of the solution decays exponentially as t → +∞. In the proofs, to obtain the blow-up and the exponential decay, we use the multiplier technique combined with a suitable Lyapunov functionals. Our results can be regarded as an extension and improvement of the corresponding results of [7], [10]- [12], [18]- [22], [28].

Preliminaries
First, we put Ω = (ρ, 1), Q T = Ω × (0, T), T > 0, and denote the usual function spaces used throughout the paper by the notations L p = L p (Ω), H m = H m (Ω). We denote the usual norm in L 2 by · and we denote · X for the norm in the Banach space X. We call X the dual space of X. We denote L p (0, T; X), 1 ≤ p ≤ ∞ the Banach space of real functions u : (0, T) → X measurable, such that u L p (0,T;X) < +∞, with and respectively. We remark that L 2 ,H 1 , H 2 are the Hilbert spaces with respect to the corresponding scalar products The norms in L 2 , H 1 and H 2 induced by the corresponding scalar products (2.3) are denoted by · 0 , · 1 and · 2 .
We then have the following lemmas.
Lemma 2.1. The following inequalities are fulfilled Lemma 2.2. The embedding H 1 0 → C 0 Ω is compact and for all v ∈ H 1 0 , we have Proofs of Lemma 2.1 and Lemma 2.2 are straightforward, so we omit the details.
Proof of Lemma 2.4. It is well known that the embedding Integrating over y from ρ to 1 to get Hence ( Lemma 2.6. There exists the Hilbert orthonormal base {w j } of the space L 2 consisting of eigenfunctions w j corresponding to eigenvalues λ j such that . . Furthermore, the sequence {w j / λ j } is also the Hilbert orthonormal base of H 1 0 with respect to the scalar product a(·, ·).
On the other hand, we also have w j satisfying the following boundary value problem (2.7) The proof of Lemma 2.6 can be found in [ [27], p.87, Theorem 7.7], with H = L 2 , and a(·, ·) as defined by (2.6).
We also note that the operator A :

On the other hand
Lemma 2.7 is complete.

The existence and uniqueness theorem
First, we make the following assumptions: 3) reduces to the following problem with the homogeneous boundary conditions Consider T * > 0 fixed, let M > 0, we put Now, for each M > 0 and T ∈ (0, T * ] , we consider the sets and we establish the linear recurrent sequence {v m } as follows.
We shall choose as first initial term v 0 ≡ṽ 0 , suppose that and associate with the problem (3.1) the following variational problem: Proof of Theorem 3.1. The proof consists of three steps.
Step 1. The Faedo -Galerkin approximation (introduced by Lions [9]). Consider the basis {w j } for H 1 0 as in where the coefficients c (k) m j satisfy the system of linear differential equations (3.11) The system (3.10) can be rewritten in form Note that by (3.6), it is not difficult to prove that the system (3.12) has a unique solution c (k) m j (t), 1 ≤ j ≤ k on interval [0, T], so let us omit the details.
Step 2. A priori estimates. We put Then, it follows from (3.10), (3.14), (3.15), that In order to estimate the terms I j we need the following lemma Lemma 3.2. We have the following estimates Proof (ii). By (3.2) 2 and (3.8) 2 , we have where It implies from (3.6) that Proof (iv). We have Proof (v). It is similarly to the proof of (iv). The proof of Lemma 3.2 is complete. Applying Lemma 3.2, we now estimate the terms I j on the right -hand side of (3.16) as follows. By Lemma 3.2 (i)-(iii) and the following inequality we obtain m (s)ds, Using integration by parts leads to

.
By Lemma 3.2 (iv) and the following inequality it is not difficult to estimate the following terms I 4 , I Hence, we deduce from (3.19) and (3.20) that Similarly, by using integration by parts from Lemma 3.2 (v) and the following inequality we also have Eq. (3.10) 1 can be rewritten as follows Hence, it follows after replacing w j withv (k) . Integrating in t to get By means of the convergences in (3.11), we can deduce the existence of a constant M > 0, it is independent of k and m such that for all m and k.

29)
and Combining  In order to get the existence and uniqueness, we shall use the following Banach space (see [9]) with respect to the norm v W 1 (T) = v L ∞ (0,T;H 1 0 ) + v L ∞ (0,T;L 2 ) . By the result given in Theorem 3.1 and by the compact imbedding theorems, we now prove the main results in this section as follows.
Taking w = w m in (3.37) 1 , after integrating in t, we get All integrals on the right -hand side of (3.39) will be estimated as below.
The integral J 1 . By (3.40), we have Hence where F M as in (3.31). Hence Using Gronwall's Lemma, we deduce from (3.46) that where k T as in (3.30).
It follows that {v m } is a Cauchy sequence in W 1 (T). Then there exists v ∈ W 1 (T) such that v m → v strongly in W 1 (T).

(3.50)
We also note that (3.51) Hence, it follows from (3.49) and (3.51) that On the other hand, we have We take w = v in (3.58) 1 and integrate in t to get it follows from (3.59) that Using Gronwall's Lemma, it follows that Z(t) ≡ 0, ie., v 1 ≡ v 2 . Therefore, Theorem 3.3 is proved.

Blow-up result
In this section, Prob.
where λ > 0, 0 < ρ < 1 are given constants andũ 0 ,ũ 1 , µ, f are given functions satisfying conditions specified later. First, we assume that (H * 2 ) µ ∈ C 1 (R + ) and there exists the constant µ * > 0 such that µ(y) ≥ µ * > 0, ∀y ∈ R + ; Then we obtain the following theorem about the existence of a weak solution. Next, in order to obtain a blow-up result in Theorem 4.2 below, we make more the following assumptions.
It is clearly that f (0) = 0. By integration by parts, we obtain Proof of Theorem 4.2. It consists of two steps, in which the Lyapunov functional L(t) is constructed in step 1 and then the blow-up is proved in step 2.

Exponential decay of solutions
This section investigates the decay of the solution of Prob.
We know that f ∈ C 1 (R) and f (0) = 0. For y ≥ 0, 1 p y f y − 2kβ p 2 y p ln k y 2 + e = p − 2k p 2 y f y .
Let µ * max = max Proof of Theorem 5.1. First, we need the following lemmas.