Abstract
In this paper, we studied a family of the exponential attractors and the inertial manifolds for a class of generalized Kirchhoff-type equations with strong dissipation term. After making appropriate assumptions for Kirchhoff stress term and nonlinear term, the existence of exponential attractor is obtained by proving the discrete squeezing property of the equation, then according to Hadamard’s graph transformation method, the spectral interval condition is proved to be true, therefore, the existence of a family of the inertial manifolds for the equation is obtained.
Highlights
In the study of dynamic behavior for a long time in infinite dimensional dynamical system, the exponential attractors and inertial manifolds play a very important role
Inertial manifold is finite dimensional invariant smooth manifolds that contain the global attractor and attract all solution orbits at an exponential rate, their corresponding inertial manifold forms are powerful tools which could study the property of finite dynamical system about the dissipative evolution equation
Can contact finite dimensional dynamical system and infinite dimensional dynamical system, many scholars have done a great deal of research, we could refer to ([2]-[8])
Summary
In the study of dynamic behavior for a long time in infinite dimensional dynamical system, the exponential attractors and inertial manifolds play a very important role. The assumption of g (u ) satisfies the following conditions: Under these reasonable assumptions, according to Hadamard’s graph transformation method, the existence of the inertial manifolds for the equation is obtained. We study the existence of exponential attractors and a family of the inertial manifolds for a class of generalized Kirchhoff-type equation with damping term:.
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