Strong Solutions and Global Attractors for Kirchhoff Type Equation

We study the long-time behavior of the Kirchhoff type equation with linear damping. We prove the existence of strong solution and the semigroup associated with the solution possesses a global attractor in the higher phase space.


Introduction
We consider the following nonlinear Kirchhoff type equation with the initial-boundary conditions: where Ω is a bounded domain in   with smooth boundary, △ denotes the Laplace operator,  is a given function lying in  2 (Ω), independent of time, and  ∈  1 () with (0) = 0 fulfills the dissipation inequality lim inf and where  > 0 is a real number and  1 is the first eigenvalue of −△ on  2 (Ω) with Dirichlet boundary conditions When  = 1, this problem describes, for instance, the motion of a vibrating string with fixed boundary in a viscous medium.In particular, the function represents the displacement from equilibrium,   is the velocity, and the term () −  may correspond to a (nonlinear) elastic force.For more details on the model of Kirchhoff, one can refer to [1][2][3] and the reference therein.
When the coefficient of   is a positive function (), which depends on , then the term ()  is a resistance force and the model impresses that the viscous medium embedding the string is stratified.In this case, the existence of the global and exponential attractors has been proven by S. Kolbasin in [4].The attractor is in the phase space  2 0 (Ω)× 2 (Ω) and it is bounded in [ 4 (Ω) ∩  2 0 (Ω)] ×  2 0 (Ω), where Ω is a bounded domain in  3 with smooth boundary.
For Ω a three-dimensional unbounded domain and under suitable conditions on () and (), A.Kh. Khanmamedov in [6] showed that this equation possesses a global attractor in  2 ( 3 ) ×  2 ( 3 ).About Kirchhoff models, long-time dynamics properties were studied by Yang and I. Chueshov in [7][8][9] and their references.For example, Yang et al. discussed the long-time behavior of solutions to the Cauchy problem of some Kirchhoff type equations with a strong dissipation in [7,8] and proved that the dynamical system possesses a global attractor under suitable conditions in the phase space  1+ , where  1+ =  1+ ×   , 0 <  ≤ 1.
Thus, to the best of our knowledge, the research about global attractors of the weak solutions for problem (1) with respect to the norm of  2 ×  2 is much more; however, the results about the existence of the strong solutions and strong global attractors for (1) are relatively fewer.The purpose of this paper is to supplement some conclusions for the above problem.In particular, we do not demand the function () to be bounded by polynomials and present here a method different from [9][10][11][12][13][14]. Furthermore, the global attractor is established in the higher energy space in  4 (Ω) ∩  2 0 (Ω).The paper is organized as follows.In Section 2, we recall some notations and some general facts about the dynamical systems theory.In Section 3, we prove well-posedness and the existence of a bounded absorbing set for problem (1).Sections 4 and 5 contain our main results, and we prove the existence of strong solution and a global attractor in the space of higher order.

Preliminaries
Throughout the paper we will denote where  = △ 2 . ( The norm and the scalar product in  2 (Ω) is denoted by ‖⋅‖ and (, ), respectively, the norm in   is denoted by ‖⋅‖   , ( = 0, 1, 2), and the Hilbert spaces  1 ,  2 are For any given function (), we will write for short () = ((),   ()) and endow space  1 ,  2 with the standard inner product and norm For convenience, the letters  and  present different positive constants and different positive increasing functions, respectively.
We collect some basic concepts and general theorems, which are important for getting our main results.We refer to [14][15][16][17][18] and the references therein for more details.
Theorem 3 (see [14]).Let X be a Banach space and {()} ≥0 be a norm-to-weak continuous semigroup on X.Then {()} ≥0 has a global attractor in X provided that the following conditions hold true: (i) {()} ≥0 has a bounded absorbing set  0 in X.
Theorem 4 (see [16,18]).Let X and Y be two Banach spaces such that  ⊂  with a continuous injection.If a function  belongs to  ∞ (0, ; ) and is weakly continuous with values in Y, then  is weakly continuous with values in X.
Theorem 5 (see [16,18]).Let X, Y be two Banach spaces and  * ,  * be their dual spaces, respectively, such that where the injection  :  →  is continuous and its adjoint,  * :  * →  * , is a densely injective.Let {()} ≥0 be a semigroup on X and Y, respectively, and be a continuous semigroup or a weak continuous semigroup on Y. Then for any bounded subset B of X, {()} ≥0 is norm-to-weak continuous on ().
Theorem 6 (see [16][17][18]).Assume that {()} ≥0 is a semigroup on Banach space X and satisfies that And assume furthermore that {()} ≥0 is norm-to-weak continuous on ( 0 ).Then {()} ≥0 has a global attractor A in X; i.e., A is nonempty, invariant, compact in X and attracts every bounded subset of X in the norm topology of X.

A Priori Estimate
Next we iterate some main results in [4], which are important for getting a priori estimate.

Existence of the Strong Solution
The corresponding eigenvalues are   ,  = 1, 2, ⋅ ⋅ ⋅ , satisfying For this purpose, according to the basis theory of ordinary differential equations, we build the sequence of Galerkin approximate solutions.They are smooth functions of the form and it satisfies where It is easy to pass the limit in (27) and we obtain that  is a solution of (1), such that  () ∈  ∞ (0, ;  ()) ,   () ∈  ∞ (0, ;  1 ) .
(32) Furthermore, from Theorem 4 and [4], we know that  is a weakly continuous function from [0, ] to  2 .Finally, uniqueness is followed from [4], since any strong solution would be a weak solution.
Thus, the dynamical system generated by ( 1) can be defined in the phase space  2 , and the corresponding solution semigroup is {()} >0 .By Theorem 7, we have the following results.
Theorem 8. Suppose the conditions of Theorem 7 hold; then there exists a bounded absorbing set  in  2 for the semigroup {()} >0 .
Similarly, we can prove the following Lemma.
Since () ×  1 →  1 ×  2 , from Theorem 5 and [4], we can immediately obtain the following result.(37) Now we give our main results of the paper.