Blow-up And Large Time Behavior For The Nonlinear Evolution Equations

The thesis deals with two types of nonlinear evolution equations:Magnetic Za-kharoy system and viscous biharmonic Camassa-Holm equations.Four chapters are included in the thesis.The first chapter is Introduction,in which the physical back-ground and research status for the above two systems are introduced.In the second chapter,we study the three-dimensional magnetic Zakharov system in a cold plasmaThe system describes physically the so-called spontaneous generation of a mag-netic field in a cold plasma.We will study finite-time singularity of radially symmetric solutions.It is a coupled system between the Schrodinger equations and a wave equa-tion with second-order derivative nonlinearity.The main difficulty to achieve the finite time blow up lies in the presence of the second-order derivative nonlinearity in the wave equation and the nonlocal term generated by the magnetic field.Motivated by the nu-merically observed results,we first explore a particular relation between E and n,that is,there exist ?>0 and C>0 such that for any t>0,there holdsWe then figure out some detailed and tricky estimates for the nonlocal terms involving rotation,vector product and Fourier transform.Finally,we prove the finite time blowup of radially symmetric solutions to the system under consideration by introducing a local auxiliary weighted function and employing limiting argument to these estimates coming from the nonlocal terms,namely,there exists T0>0 such thatIn the third chapter,we tackle large time behavior and convergence for the n-dimensional(n=2,3)viscous biharmonic Camassa-Holm equations in the entire space:By employing some properties of Fourier transform,we first establish the non-uniform decay.We then derive the algebraic decay by Fourier splitting method.Finally,we show the convergence from the biharmonic Camassa-Holm equations to the corresponding Navier-Stokes equations in a suitable norm space as(?,?)?(0,0)by applying some estimates in elliptic theory.