Long-time Dynamics For Nonlinear Berger Equation With Infinite Memory

During the high-speed flight of an aircraft,the surface of the fuselage often encounters unstable airflow and generates non-linear oscillation.Berger equations are described such problems,and they have attracted widespread attention from experts and scholars at home and abroad.We consider the long-time dynamics behavior of Berger equations with memory terms.By constructing a functional equivalent to the energy function,it is proved that the dynamic system produced by the solution of the equation is a gradient system with quasi-stable properties,so the existence of a compact global attractor with a finite fractal dimension is obtained.On this basis,it is further proved that the regularity of the attractor with respect to time and the existence of a finite-dimensional generalized exponential attractor.The paper has been divided into four sections:The first section: We elaborate the research background of Berger type equations and the current research status of Berger equations at home and abroad and introduces the main research work and results of the equation.The second section: We present basic concepts of dynamical systems and gives the inequalities and related lemmas required for proof.The third section: We introduce new variables to process the memory term.The nonautonomous dynamical system is transformed into autonomous.The well-posedness of the solution of Berger equation with memory are proved according to the semigroup theory.The solution mapping of Berger type equation with memory term yields a dynamic system.The fourth section: We study the long-term dynamic behavior of the dynamic system.Since the introduction of the memory term weakens the strong damping,it is difficult to prove the existence of the absorbing set,and it is also difficult to verify the compactness of the semigroup if the usual methods are used.Therefore,applying the new technology proposed by Chueschov and other scholars,the gradient properties are obtained by proving the existence of strict Lyapunov functions in the dynamic system.Then constructing a functional equivalent to the energy function proves that the dynamic system has quasi-stable properties,and thus obtains a compact global attractor with a finite fractal dimension.