Non-existence of solutions for a Timoshenko equations with weak dissipation

In this paper, we consider the following Timoshenko equation utt +4u−M ( ‖∇u‖ ) 4 u+ ut = |u| u associated with initial and Dirichlet boundary conditions. We prove the non-existence of solutions with positive and negative initial energy.

In the case of M (s) = 1 and without fourth order term 2 u, the equation ( 1) can be written in the following form The existence and blow up in finite time of solutions for (2) were established in [6,7,8,10,15].
In the case of M (s) = 0 the equation ( 1) can be written in the following form Messaoudi [11] studied the local existence and blow up of the solution to the equation (3).Wu and Tsai [16] obtained global existence and blow up of the solution of the problem (3).Later, Chen and Zhou [2] studied blow up of the solution of the problem (3) for positive initial energy.
The problem (1) was studied by Esquivel-Avila [4,5], he proved blow up, unboundedness, convergence and global attractor.Pişkin [12] studied the local and global existence, asymptotic behavior and blow up.Later, Pişkin and Irkıl [13] studied blow up of the solutions (1) with positive initial energy.
In this paper, we prove the nonexistence of solutions for the problem (1), with positive and negative initial energy.
This paper is organized as follows.In section 2, we present some lemmas and notations needed later of this paper.In section 3, nonexistence of the solution is discussed.

Preliminaries
In this section, we shall give some assumptions and lemmas which will be used throughout this paper.Let .and .p denote the usual L 2 (Ω) norm and L p (Ω) norm, respectively.Lemma 2.1 (Sobolev-Poincare inequality [1]).Let p be a number with 2 ≤ p < ∞ (n = 1, 2) or 2 ≤ p ≤ 2n n−2 (n ≥ 3) , then there is a constant We define the energy function as follows ( 4) Proof.Multiplying the equation of (1) by u t and integrating over Ω, using integrating by parts, we get Next, we state the local existence theorem of problem (1), whose proof can be found in [12].
Then there exists a unique solution u of (1) satisfying . Moreover, at least one of the following statements holds:

Non-existence of solutions
In this section, we deal with the blow up of the solution for the problem (1).Let us begin by stating the following two lemmas,which will be used later.
4 Non-existence of solutions for a Timoshenko equations with weak. . .Definition 3.1.A solution u of ( 1) is called blow up if there exists a finite time T * such that , and that γ ≥ 0, then we have Proof.By differentiating (6) with respect to t, we have Then from ( 4) and ( 9), we have Since γ 2 ≤ δ ≤ q−1 4 , we obtain (7).
Theorem 3.1.Assume γ 2 ≤ δ ≤ q−1 4 , γ ≥ 0 and one of the following statements are satisfied Then the solution u blow up in finite time T * in the case of (5).
In case (i) Furthermore, if H (t 0 ) < min 1, − a b , we have , where In case (ii) In case (iii) where a and b are given ( 16), (17). Proof.Let where T 1 > 0 is a certain constant which will be specified later.Then we get and where For simplicity of calculation, we define From ( 8), (12) and Hölder inequality, we get ( 21) If case (i) or (ii) holds, by (7) we have Thus, from (20)-( 22) and (18), we obtain where By the Schwarz inequality, and Θ (t) being nonnegative, we have