Dynamics For Some Classes Of Nonlinear PDES

In this thesis,by using perturbation theory,weakly nonlinear analysis,center manifold theory and normal form methods,we study the dynamics of some classes of nonlinear partial differential equations.Chapter 1 is concerned with the asymptotic behavior of the solution u(x,t)of a particular system of parabolic partial differential equations as amplitude equations on a bounded domain(0,L)with Neumann boundary conditions.For the stationary solution(0,c),with the length L of the domain regarded as bifurcation parameter,asymptotic expression of the nontrivial solutions bifurcated from(0,c)are obtained by using perturbation method.Moreover,the stability of the nontrivial solutions is discussed.On the other hand,for the steady state(c,1-c2)(c≠0),by employ-ing weakly nonlinear analysis,we obtained a pitchfork bifurcation bifurcated from the steady state(c,1-c2)(c,0),and the direction of the pitchfork bifurcation is discussed.In Chapter 2,we study the bifurcation and stability of trivial stationary solution(0,0)of coupled Kuramoto-Sivanshinsky and Ginzburg-Landau-type equations(KS-GL)on a bounded domain(0,L)with Neumann boundary conditions.The asymptotic behavior of the trivial solution of the equations is considered.For there were four or-der derivatives in the second formula,we cannot express one variable in the form of the other variable from the second formula of the steady state eqations,thus the discus-sion will be more complicated than that of Chapter 1.Using the perturbation method,some different cases are discussed,and asymptotic expressions of the nontrivial solu-tions bifurcated from the trivial solution are obtained.Moreover,the stability of the bifurcated solutions is discussed,then the range of parameter which guarantees the stability of the nontrivial solution is obtained.In Chapter 3,we consider the Hopf bifurcation and the Turing instability of the reaction-diffusion Gierer-Meinhard model.First we consider the corresponding ordi-nary differential equations of the Gierer-Meinhard model and obtain the stability of the equilibrium solution and the existence,stability and directions of the Hopf bifur-cation.After that,we discuss the Turing instability of the equilibrium solution which is both the solution of the ODE and PDE systems and find that the stable equilibrium solution becomes unstable under some parameter range because of diffusion.At last,we discuss the stability and direction of the Hopf bifurcation of the PDE system.We deduce that under some parameter range,the homogeneous stable periodic solutions bifurcated from the Hopf bifurcation point will become unstable because of diffusion.In Chapter 4,we consider the shadow system of the Gierer-Meinhard model.The existence,stability and bifurcation of steady solution for the system are discussed.We study the solution by using nonlinear functional analysis,Jacobi elliptic integrals and bifurcation theory,and obtained the range of parameter which guarantees the existence and stability of the steady solution.Moreover,under some conditions,a Hopf bifurcation occurs at some critical point.