Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms
Jian Dang, Qingying Hu^{*}, Hongwei Zhang
Department of Mathematics, Henan University of Technology, Zhengzhou, China
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To cite this article:
Jian Dang, Qingying Hu, Hongwei Zhang. Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms. International Journal of Applied Mathematics and Theoretical Physics. Vol. 2, No. 4, 2016, pp. 41-45. doi: 10.11648/j.ijamtp.20160204.13
Received: August 21, 2016; Accepted: August 29, 2016; Published: October 14, 2016
Abstract: In this paper, we are concerned with one-dimensional Timoshenko model for a beam with nonlinear damping and source terms. We establish a blow-up result when the initial energy is positive and the initial data is not in a potential well. Under arbitrary positive initial energy, we also prove a finite-time blow-up result for a special case.
Keywords: Timoshenko System, Source Term, Damping Term, Blow-up
1. Introduction
In this paper, we study the semilinear Timoshenko system
(1)
(2)
in , under the following boundary conditions
(3)
(4)
and initial conditions
(5)
(6)
The function is the rotation angle of a filament of the beam and is the transverse displacement of the beam, is a strict positive constant. The nonlinear function and act as strong source terms, . For the corresponding linearized system of (1)-(2)
(7)
(8)
which is given by Timoshenko [1] as a simple model describing vibration of a beam, this model for Timoshenko beams have attracted vast interest during the last thirty years. Systems (7)-(8) has been studied by many authors and results concerning existence and asymptotic behavior have been established. The stabilization of the Timoshenko system has been studied with different type of damping, we refer the reader to [2,3,4,5,6] and their references.
Let us mention some known results for semilinear Timoshenko system. Parente et al [7] treated the existence and uniqueness for the problem
(9)
(10)
with differential boundary conditions, where are Lipschitz continuous functions. Araruna et al [8] investigated the existence and uniqueness of strong and weak solution of the one-dimensional Timoshenko model (9)-(10) for beams with a nonlinear external forces and a boundary damping mechanics. They also proved that the energy of solution decays exponentially. Chueshov and Lasiecka [9] studied the existence of a compact global attractor for (9)-(10) with nonlinearities of and being locally Lipschitz in the 2-dimensional case. Gorgi and Vegni [10] gave the uniform energy estimate and the estimate of an absorbing set for the Timoshenko beam with memory and Dirichlet boundary condition. Messaoudi and Soufyane [6, 11] established a general decay result for a nonlinear Timoshenko system with a boundary control of memory type
(11)
(12)
However, there has been less focus on the Timoshenko system with nonlinear source terms. Recently, Pei et al [12,13] studied the global well-posedness and long-term behavior of the Reissner Mindlin-Timoshenko plate systems, focusing on the interplay between nonlinear viscous damping and source terms, by the potential well framework [14,15]. To the best of our knowledge, the system of nonlinear Timoshenko equation have not been well studied.
In this paper, we consider the blowup of the solutions of problem (1)-(6). We give an equivalent inequality between
and the standard norm on the function space , then we obtain local existence of solution of problem (1)-(6) following very carefully the techniques used in [16]. We prove that the solutions blow up in finite time if when the initial energy is positive and the initial data is not in a potential well. The main tool of the proof is a technique introduced by paper [17] and some estimates used firstly by Vitillaro [18], in studying a class of a single wave equation. This tool has been used by many paper to deal with the global existence and blow-up of solutions to some nonlinear hyperbolic systems with damping and source terms in a bounded domain, for example see [19, 20]. Secondly, we extend the result of [21], established for the Klein-Gordon equation, to our problem, and prove a finite-time blow-up result for for problem (1)-(6) under arbitrary positive initial energy.
2. Preliminaries
Throughout this paper, we denote and by and , respectively. and denote the usual norm andnorm, respectively. And let us define
as the usual inner product. The standard duality between and will be denote also by . For , sometimes we write equivalent norm instead of norm
Let V denotes the following Hilbert space and endowed with the following norm for . Throughout this paper, are positive generic constants, which may be different in various occurrences. In addition, we denote is the Poincare constants, that is, for
(13)
Concerning the nonlinear functions and , we assume that
(14)
(15)
where and are constants. It is easy to see that
(16)
for any , where
Moreover, a quick computation will show that there exist two positive constants and such that the following inequality holds (see [19, 20])
(17)
Now, we give the following lemmas which will be used later. By a simple computation, we have the following result:
Lemma 2.1 For , there exist positive constants such that the inequality holds
(18)
where
.
Now, we state the local existence of the problem (1)-(6) and the proof follows very carefully the techniques used in [16].
Lemma 2.2 (Local existence) Assume that the assumptions (14)-(18) hold. Suppose further that , then for any initial data and , there exists a local weak solution of problem (1)-(6) defined in for some and satisfies the energy identity
(19)
where is defined by
(20)
And
(21)
It follows from Theorem 2.2 that
(22)
then the energy function is a nonincreasing function.
Lemma 2.3 There exist such that for any , the following inequality holds
(23)
Proof A combination of the following inequality [17]
with Lemma 2.1 yields (23).
3. Blow-up of Solution
In this section, we deal with the blow-up of the solution of problem (1)-(6). Firstly, based on the technique introduces by paper [17] and some estimates used firstly by Vitillaro [18], in studying a class of single wave equation, we establish a blow-up result when the initial energy is positive and the initial data is not in a potential well. Secondly, we extend the result of [21], established for the Klein-Gordon equation, to our problem. Under arbitrary positive initial energy, we prove a finite-time blow-up result for for problem (1)-(6).
For the sake of simplicity, we set . Let
, ,
(24)
where is the optimal constant in (23).
Lemma 3.1 Let be a solution of problem (1)-(6), , and
(25)
then there exist a constant , such that
(26)
(27)
Proof. By the definition of, (14)-(18), and the definition of, we have
(28)
where . Then we can get the result by using the proof of Lemma 3.3 in [17] word for word.
Now, we state the result of blow-up of solution.
Theorem 3.2 Assume that
,
then any solution of problem (1)-(6) with initial data satisfying
and , cannot exist for all time.
The proof is similar to that in [17], so we give only the result here.
Now, we prove a finite-time blow-up result for for problem (1)-(6) under arbitrary positive initial energy.
Lemma 3.3 [21] Suppose that satisfies
(29)
and
(30)
Then, there exist such that
and the estimate of the life span of the solution is given by , where
(31)
Theorem 3.4 Assume that in equations (1) and (2). If
where
Then there exist , such that , where is defined in (31).
Proof We first multiply both sides of equation (1) and (2) by and, respectively, and integrate over, then summing up and integrating by parts, we obtain the following equality
(32)
where .
Similarly, multiplying both sides of equation (1) and (2) by and, respectively, summing up and integrating by parts, we obtain the following equality
(33)
Integrating (33) over and noting , we arrive that
,
and then, combining the expression of , we have
(34)
By (17), combining (32) and (34), we get
(35)
Thus (35) yields the following differential inequality
(36)
From Cauchy-Schwartz inequality, we have
(37)
Multiplying (36) by (t) and using (37), we obtain
(38)
Comparing this differential inequality with (29), we easily see that for
as in (30), the time is bounded above by
,
(39)
For
,
.
Thus we have the desired results.
Acknowledgements
This work is supported by National Natural Science Foundation of China (No. 11526077) and Basic Research Foundation of Henan University of Technology (171164).
References