The Study Of The Blow-up Solution And The Attractor To The Generalized Hyperelastic-rod Equation

The hyperelastic-rod equation is obtained in the study of compressible elastic material. It is widely used in the areas of elasticity and has important research value. In this paper, we study the blow-up of solution of the Cauchy problem and the attractor of the initial boundary problem for the generalized hyperelastic-rod equation. There are four sections in this paper.The first section, we introduce the background, actuality and summarize the main results.The second section, we introduce the basic concepts, the basic lemmas and several important inequalities.The third section, we consider the Cauchy problem of the generalized hyperelastic-rod equation. Using a prior estimate from the perspective of characteristic curve, we get the blow-up of solution and its exact blow-up rate in a limited period of time under certain conditions.The fourth section, we discuss the dynamics of the initial boundary problem for a class of generalized hyperelastic-rod equation. The existence of global solution in H~3 is gained under certain conditions using some a prior estimates and Galerkin method, and finally the existence of absorbing set and the existence of global attractor in H~2 are obtained in this section.