Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms

Abstract We consider the existence, both locally and globally in time, the decay and the blow up of the solution for the extensible beam equation with nonlinear damping and source terms. We prove the existence of the solution by Banach contraction mapping principle. The decay estimates of the solution are proved by using Nakao’s inequality. Moreover, under suitable conditions on the initial datum, we prove that the solution blow up in finite time.


Introduction
We consider the initial-boundary value problem for the following extensible beam equation with nonlinear damping and source terms u .x; 0/ D u 0 .x/ ; u t .x; 0/ D u 1 .x/ ; x 2 ; u .x; t/ D @ @ u .x; t/ D 0; x 2 @ ; (1) where p; q 1 are real numbers, is a bounded domain with smooth boundary @ in R n ; is the outer normal, and M .s/ is a positive locally Lipschitz function as M .s/ D˛Cˇs ;˛;ˇ 0; 1: This kind of wave equation is obtained from the extensible beam equation of Woinowsky-Krieger [1], for g D 0; where u .x; t/ is the deflection of the point x of the beam at the time t and˛1;ˇ1 > 0 are constants. The equation (2) was studied by many authors such as [2][3][4][5][6][7][8].
In the case of M .s/ D 1 and without fourth order term 4 2 u; the equation (1) can be written in the following form u t t 4u C ju t j p 1 u t D juj q 1 u: The existence and blow up in finite time of solutions for (3) were established in [9][10][11][12][13]. The interaction between the damping .ju t j p 1 u t / and the source term .juj q 1 u/ make the problem more interesting. Levine [10,11] first considered the interaction between the linear damping .p D 1/; and source term by using the Concavity method. But *Corresponding Author: Erhan Pişkin: Dicle University, Department of Mathematics, 21280 Diyarbakır, Turkey, E-mail: episkin@dicle.edu.tr this method can not be applied to the case of nonlinear damping term. Georgiev and Todorova [9] extended Levine's result to the nonlinear case .p > 1/ : Recently, the problem (1) was studied by Esquivel-Avila [14,15], he proved blow up, unboundedness, convergence and global attractor.
In this paper, we analyze the influence of the damping terms and source terms of the solution of the problem (1). We prove the local existence of solutions for the problem (1) by Banach contraction mapping principle. After that, we obtained global existence, decay and blow up of solutions.
This paper is organized as follows. In Section 2, we present some lemmas and notations needed later in this article. In Section 3, we prove the local existence for the problem (1). The proof of the global existence and decay of the solution are given in Section 4. In Section 5, blow up of the solution is discussed.

Preliminaries
In this section, we shall give some assumptions and lemmas which will be used throughout this paper. Let k:k and k:k p denote the usual L 2 . / norm and L p . / norm, respectively. Now, we state the general hypotheses (H1) M .s/ is a positive locally Lipschitz function for s 0 with the Lipschitz constant L satisfying M .s/ m 0 > 0: (H2) For the nonlinearity, we suppose that 1 < q < 1 if n Ä 2; and 1 < q Ä n n 2 if n > 2: Lemma 2.1 (Sobolev-Poincare inequality [16]). Let p be a number with 2 Ä p < 1 .n D 1; 2/ or 2 Ä p Ä 2n n 2 .n 3/ ; then there is a constant C such that kuk p Ä C kruk for u 2 H 1 0 . / : 17]). Let .t / be nonincreasing and nonnegative function defined on OE0; T ; T > 1; satisfying for w 0 is a positive constant and˛is a nonnegative constant. Then we have, for each t 2 OE0; T ; .t/ Ä .0/ e w 1 OEt 1 C ;˛D 0; .

Local existence
In this section, we are going to consider the local existence of the solution for the problem (1) by the similar arguments as in [18].
Theorem 3.1. Assume that (H1) and (H2) hold, and that .u 0 ; u 1 / 2 H 2 0 . / L 2 . / ; then there exists a unique solution u of (1) satisfying .0; T // : Moreover, at least one of the following statements holds: Proof. Define the following two parameter space for T > 0 and R 0 > 0: Then X T;R 0 is a complete metric space with the distance d .
We define the nonlinear mapping S in the following way. For v 2 X T;R 0 ; u D S v is the unique solution of the following equation

<
: u .x; 0/ D u 0 .x/ ; u t .x; 0/ D u 1 .x/ ; x 2 ; u .x; t/ D @ @ u .x; t/ D 0; x 2 @ : We shall show that there exist T > 0 and R 0 > 0; such that (i) S W X T;R 0 ! X T;R 0 ; (ii) S is a contraction mapping in X T;R 0 with respect to the metric d .:; :/ : First, we shall check (i), then the existence of the solution u for (7) will be proved by a standard method. Multiplying (7) by 2u t and integrating over .0; t/ ; we obtain where We calculate that where L is a Lipschitz coefficient. Also by Hölder inequality and Sobolev-Poincare inequality, we get Thus, (10) and (11) into (8), we obtain Integrating (12) over OE0; T and by Gronwall inequality, we have (6) and (13), we obtain From (14), we have which implies u 2 C OE0; T / I H 2 0 . / : In order to prove that the map S satisfies (i), it will be enough to show that the parameters T and R 0 satisfy In the following, we show that u t 2 C OE0; T / I L 2 . / : For each t 0 2 OE0; T ; let Multiplying (16) by 2w t and integrating over .0; t/ ; we obtain where (17) over OEt 0 ; t and using the fact e 2 .w .t 0 // D 0; we obtain We observe from (15), that and Hence, by the monotonicity of ju t j p 1 u t ; when t > t 0 ; we see On the other hand, when t < t 0 ; from the fact that u t 2 L pC1 ..0; T / / ; we have On the other hand, we shall check (ii), we take v 1 ; v 2 2 X T;R 0 and denote u 1 D S v 1 and u 2 D S v 2 : Hereafter we suppose that (15) holds, so that u 1 ; u 2 2 X T;R 0 : when we put w D u 1 u 2 ; w satisfies the following: Multiplying (18) by 2w t and integrating over .0; t/ ; we obtain d dt To proceed the estimates of I 6 ; I 7 and I 8 ; we observe that

t //
and Then, integrating (20) over .0; t/ and using initial-boundary conditions (18), we have Then, by definition d .u; v/, we have we can see S is a contraction mapping. Finally, we choose suitable T and R 0 small so that (i) and (ii) hold. By Banach contraction mapping theorem, we obtain the local existence.
The second statement of the theorem is proved by a standard continuation argument (see [19]). The proof of the theorem is now completed.

Global existence and decay of solutions
In this section, we discuss the global existence and decay of the solution for the problem (1). In this section and next section, we take M .s/ D˛Cˇs ;˛;ˇ 0; 1: Obviously M .s/ satisfies Lipschitz condition. Without loss of generality, we can assume that˛DˇD 1: We define and We also define the energy function as follows The next lemma shows that our energy functional (23) is a nonincreasing function along the solution of (1). Proof. Since I .0/ > 0; it follows the continuity of u .t / that for some interval near t D 0: Let T m > 0 be a maximal time, when (28) holds on OE0; T m : From (21) and (22), we have By using (29), (23), and Lemma 4.1, we obtain By recalling Sobolev-Poincare inequality, (30) and (27), we have Therefore, by using (22), we conclude that I .t/ > 0 for all t 2 OE0; T m : By repeating the procedure, T m is extended to T: The proof of Lemma 4.2 is completed. Proof. It is sufficient to show that ku t k 2 C k4uk 2 C kruk 2 C kruk 2. C1/ is bounded independently of t: In order to achieve this we use (22) and (23), we obtain where w 1 ;˛and C 9 are positive constants which will be defined later.

Blow up of solutions
In this section, we are going to consider the blow up of the solution for problem (1). Blow up of the solution with negative initial energy was proved for q > max f2 C 1; pg by using the technique of [9].
where " small to be chosen later and 0 < Ä min q p p .q C 1/ ; q 1 2 .q C 1/ : Our goal is to show that ‰ .t/ satisfies a differential inequality of the form This, of course, will lead to a blow up in finite time.
By taking a derivative of (49) and using Eq. (1) we obtain By using the definition of the H .t/ ; it follows that Inserting (52) into (51), we obtain In order to estimate the last term in (53), we make use of the following Young inequality where X; Y 0; ı > 0; k; l 2 R C such that 1 k C 1 l D 1: Consequently, applying the previous we have where ı is constant depending on the time t and specified later. Therefore, (53) becomes Therefore, by taking ı so that ı pC1 p D kH .t / ; where k > 0 is specified later, we obtain Since q > p and H .t/ Ä 1 qC1 kuk qC1 qC1 ; we obtain By combining of (57) and (60), we arrive ‰ 0 .t/ ‰ 1 1