Bifurcation In Several Kinds Of Reaction-Diffusion Equation

Bifurcation theory which has rich actual background,has become one of the central issues in nonlinear analysis.meanwhile.it is also an important issue in the study of partial differential equations.The reason why bifurcation theory is important is not only its value in pure mathematics,but also its importance in applications.The researches about bifurcation especially receive more and more attentions.Many scholars study bifurcation,and have some beautiful results.In this paper,we study the bifurcation theory of reaction diffusion equation,we give a rigorous bifurcation analysis for a reactor model with combination of quadratic and cubic steps.We extend the analysis of Satnoianu et al in two directions.First,we impose Dirichlet boundary conditions in stead of the semi-infinite domain in one space dimension.Second,we consider a variety of different bifurcation phenomena of the corresponding steady state system,focusing on their parametric sensitivity.We give the detail analysis and show that the system exhibits subcritical pitchfork bifurcation, supercritical pitchfork bifurcation and transcritical bifurcation in the different regimes of control parameters,we also give the condition of the occurrence of bifurcation from the trivial solution,and consider the bifurcation directions.To compare the decades old work by Auchmuty et al,we show that the spatial structures formed by the parameter p are richer than those formed by the purely quadratic step for p=0 and the purely cubic step for p=1 respectively.Then,we study the bifurcation of traveling-wave solutions which are periodic in space about Glycolysis model.some recently considerable work has been done in describing wave-like solutions in problems with rotational symmetry.Here,we choose c as a bifurcation parameter,mainly proof transversality condition,such that we discuss the bifurcation problem of traveling-wave solution.At last,we show the result of global branching such solutions.