The Stability And Stabilizability Of Some Classes Of Parabolic Systems

This dissertation investigates mainly the stability and stabilizability of some classes of parabolic systems.For the heat equation in the non-cylindrical domain and the heat equation with time-varying coefficient,we study the stability and the rapid exponential stabilization by the boundary feedback control.For the degenerate equation in the non-cylindrical domain,we merely study the stability.For the heat equation with memory,we study the stability and the exponential stabilization of the system.There are four parts in this dissertation,every part is an independent chapter.In the first part of this dissertation(Chapter 2),we are devoted to the study of the stability and stabilizability of heat equation in non-cylindrical domain.The special solutions are given by the undetermined function method and the similarity variation.The bound estimation of special solutions shows that the system with certain boundary curve is not(analogously)exponentially stable.Then,the stability of the system is obtained by the energy estimate and the comparison principle.At last,the rapid exponential stabilization of the one dimensional system is proved by the backstepping method.In the second part of this dissertation(Chapter 3),we are concerned with the sta-bilization of two classes of systems with time-varying coefficients.We first establish the non-exponential stability of the systems,and then get the rapid exponential stabilization of the system by the backstepping method.In the third part of this dissertation(Chapter 4),we investigate the stability of degener-ate heat equation in non-cylindrical domain.For the degenerate heat equation in cylindrical domain,we obtain the exponential stability by the weighted Hardy inequality and the rapid exponential stabilization by the method of lifting the boundary.Based on the above results,we analyze the stability of degenerate heat equation in non-cylindrical domain according to the degeneration index a.The positive impact of the degeneracy on stability for degener-ate heat equation in non-cylindrical domain can be seen by comparing the stability for the system of degeneracy with that without degeneracy.The fourth part of this dissertation(Chapter 5)is addressed to a study of the stability for heat equation with memory.The system does not decay for the positive kernel,and is polynomially stable but not exponentially for the negative kernel.In particular,a more detailed description of the decay property for the negative kernel is obtained by dividing initial values into two categories:one makes the solution exponentially stable and the other makes the solution only polynomially stable.Moreover,the exponential stabilization is obtained by the method of boundary homogenization.