Abstract
We investigate the global well-posedness and the longtime dynamics of solutions for the higher-order Kirchhoff-typeequation with nonlinear strongly dissipation:2( ) ( )m mt t tu  ï€ ï„ u  ï¦ ï D u ï ( ) ( ) ( )mï€ ï„ u  g u  f x . Under of the properassume, the main results are that existence and uniqueness of the solution is proved by using priori estimate and Galerkinmethod, the existence of the global attractor with finite-dimension, and estimation Hausdorff and fractal dimensions of theglobal attractor.
Highlights
We consider the problem m u tt m ( ) ut ( ) u m )( ) u g (u )f ( x ), x, t> 0,m > 1 (1.1) u ( x, t ) iu vi0, i = 1, 2, m - 1, x, t>0 (1.2)
We investigate the global well-posedness and the longtime dynamics of solutions for the higher-order Kirchhoff-type equation with nonlinear strongly dissipation: u tt
Zhijian Yang and Pengyan Ding [8] studied the longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on R n ) L2 ( (R n) : u tt u t ( u 2 ) u u t g ( x, u ) f ( x )
Summary
Where is a bounded domain of R n , with a smooth Dirichlet boundary and initial value, the damping coefficient is function of the L2-norm of the gradient m power, g(u) is a nonlinear forcing, ( ) m u t is a strongly dissipation. Zhijian Yang and Pengyan Ding [8] studied the longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on R n :. Chueshov [10] first studied the well-posedness and the global attractor for the IBVP of Kirchhoff wave models with strong nonlinear damping:. He established a finite-dimensional global attractor in the sense of partially strong topology. Under the existence of the global solution, it is discussed that the global attractor and infinte Hausdorff dimension and fractional dimension. The main details of this paper are arranged as follow: In section 2, under the assume of Lemma 2.1 and Lemma 2.2, we get the existence and uniqueness of solution; in section 3, we obtain the global attractor of the problems (1.1)-(1.3); in section 4, we consider the finite Hausdorff dimension and fractal dimension of the global attractor
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