The existence conditions for global exponential attractor of non-autonomous evolution equations and applications

Global exponential attractor theory is an effective tool for studying the dynamic of dissipative evolution equations. However, the existence of global exponential attractor for non-autonomous dynamic system is not only a difficult problem, but also an unsolved problem. In this paper, according to the structure of evolution equations, we propose two equivalent conditions for the existence of global exponential attractor of non-autonomous evolution equations. Using these two equivalent conditions, we prove the existence of bounded absorbing set, the existence of global exponentially attracting set for non-autonomous dissipative evolution equations, and study the finite fractal dimension estimation for the global exponentially attracting set of non-autonomous dissipative evolution equations. Finally, as applications, the existence of global exponential attractor is proved for non-autonomous semilinear parabolic equation and the epidemic model. Our results also provide a convenient and effective tool to study the global exponential attractor.