Dimension Of The Attractor And Kernel Sections For Damped Wave Equations With Critically Growth Exponent

Infinite-dimensional dynamical systems play an important role in nonlinear science, nonlinear wave equation is a kind of very important infinite-dimensional dynamical systems. The research on the attractor lies in two aspects. one aspect is the existence of the attractor, the other is the geometry properties of the attractor under the existence of the attractor, such as Hausdorff dimension.This paper firstly introduce the development survey and main research directions of dynamical systems and preliminary results and definitions, the basic function spaces and frequently used inequalities such as Young's Inequality, Holder's Inequality and Gronwall's Inequality, then, the author briefly introduce the research work of this paper.The research work of this paper consist of three chapter.In chapter 2, we discuss the semilinear wave equation when there is a nonlinear damping and the nonlinearity satisfies the critical growth exponent, we prove that the semigroup of the equation can produce a global attractor. In chapter 3, we consider the estimate of the upper bound of Hausdorff dimension of the global attractor for strongly damped semilinear wave equation with critical growth exponent. By using the boundedness of the global attractor in high regularity space, we optimize the estimate of the upper bound of Hausdorff dimension of the global attractor with non-critical growth exponent, and show that it is also the estimate of the upper bound of Hausdorff dimension of the global attractor with critical growth exponent. In chapter 4, we consider a kind of non-autonomus wave equations with critical growth exponent, using the system with the as-ymptoticly regular solution, we obtain its kernel sections are also asymptoticly regular, moreover, we show the estimate of Hausdorff dimension of kernel sections.