Global Existence , Uniqueness of Weak Solutions and Determining Functionals for Nonlinear Wave Equations

We consider the initial-boundary value problem for a nonlinear wave equation with strong structural damping and nonlinear source terms in IR. We prove the global existence and uniqueness of weak solutions of the problem and then we will study the determining modes on the phase space     2 1 0 0,1 0,1 H H  by using energy methods and the concept of the completeness defect.


Introduction
In this paper we study the initial-boundary value problem for the following nonlinear wave equation 3) was studied in [1][2][3][4].In [1] Chen et al worked that the following initial boundary value problem     0, 1, 0, > 0 u t u t t   (1.6) has a global solution and there exists a compact global attractor with finite dimension.In [2] Karachalios and Staurakalis studied the local existence for (1.1) with 0,   u t is a damping term and without nonlinear source term.In [3] Çelebi and Uğurlu gave the existence of a wide collection of finite sets of functionals on the phase space    2 1 0 0,1 0,1 H H   that completely deter- mines asymptotic behavior of solutions to the strongly damped nonlinear wave equations.In [4] Chueshov presented the approach of a set of determining functionals containing determining modes and nodes that completely determines the long-time behavior of some first and second order evolution equations.
Similar results of determining modes for similar equations have been obtained in [5][6][7].
In this article, we take the problem defined by (1.1)-(1.3)which was not investigated in above mentioned articles.Our problem has nonlinear strain and source terms.The control of long time behavior is achieved due to the presence of restoring forces  2 .


we prove the global existence and uniqueness of a weak solution u of the problems (1.1)- (1.3).In Section 3 we study determining modes on the phase space by using energy methods and the concept of the completeness defect.

The Global Existence and Uniqueness of Weak Solutions
Let be the usual Hilbert space of square integrable functions with the standard norm   is a bounded linear operator defined in see [8].The nonlinear source term where Finally we denote , .
Then the following Lemma1 is valid [9].
where   2 0,1 .f L  Using the Sobolev embedding theorem, we can see that G  is locally Lipschitz continuous.Thus we apply the existence theorem in [8] to get the solutions of initial value problem for the following system in Y:  we give the following Lemma 3. In the proofs of Lemma 3 and Theorem 4 (Global Existence) we repeat a similar technique used in [1].
where  is a constant to be determined.Thus (1.1) becomes Taking the inner product of both sides of (2.6) with v and integrating the resulting equation, we have where , , 4 4 and , 1 , .

Theorem 4 (Global Existence)
In Lemma 3 we find that Then we multiply both sides of (2.17) by  and add to (2.7) to obtain Using Poincaré inequality and (2.10) in (2.18), we h where .
Then thanks to Young inequality we obtain and we have 0 , 0 Consequently the differential form of Gronwall's inequality implies to give on Now we give some definitions, theorems and corollary

Defini
[4] L be a finite set of linear continuou

Existence of Determining Functionals
for proving existence of determining functionals.
We will say that  is a set of determining functionals for (1.1)-(1.3)when for any two solutions Definition 7 [4] Let V and H be the reflexive spaces and V be continuously and densely embedded into ves the spectral characterization of the completeness defect in The following assertion gi the case when V and H are the Hilbert spaces.
Theorem 8 [4] Let V and H be the separable H spaces such that V is compactly and densely embedded in ilbert to H. Let K be the self-adjoint, positive and compact operator in the space V defined by the equality where P  r i V on the s the orthoprojector in the space annihilato is the maximal eigenvalue of the operator S. Coro ry 9 [4] Let the conditions of Theorem 8 be hold and let us denote by   the orthonormal basis in the space V i e of that consists the eigenvectors of the operator K:   , , , Then the completeness defect of the set of functionals, Copyright © 2013 SciRes.APM can be evaluated by the formula The following theorem establishes a relation between the completeness defect and the set Theorem 10 [4] Let be the comctionals on V sitive constant such that  linear fun ists a po pleteness defect of a set  of with respect to H. Then there ex where  is the c e los d al The following version of Gronwall's lemma is also ded to determ nee ine behavior of solutions as Lemma 11 [4]

 be a locally integrab valued function on
satisfying for some the (3.6)where Theorem 12 We assume that for some and any small v, v 1 , (3.12) where C is independent of v, v 1 , v 2 [10] where are positive constants and denote the completeness defect between  and and that is From Theorem 10 we have   as small as possible so that  

Acknowledgements
The author thanks Professor A. Okay Çelebi for valuable or strain term.An other version of problems (1.1)-(1. c cerning existence of a set of determining functionals of ons to problems (1.1)-(1.3). ,