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The new bifurcation method for periodic solutions of time-dependent partial differential equations
Abstract
This dissertation is concerned with applying the abstract properties characterizing the infinite-dimensional fold and cusp singularities of maps on Banach spaces to the study of the bifurcation phenomena for time periodic solutions of nonlinear reaction diffusion equation and nonlinear telegraph equation spatial with Dirichlet boundary conditions: (UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\eqalign{u\sb{t}+Lu+f(u)&=g(x,t)\quad {\rm in}\ \Omega\times R\cr u&=0\quad {\rm on}\ \partial\Omega\times R\cr}\rm and\eqalign{u\sb{tt}+\alpha u\sb{t}+Lu+f(u)&=g(x,t)\quad {\rm in}\ \Omega\times R\cr u&=0\quad {\rm on}\ \partial\Omega\times R\cr}$$(TABLE/EQUATION ENDS)obtained by function g which is periodic in t. Here $\Omega$ is a bounded domain in $R\sp{n}$ with smooth boundary $\partial\Omega$ and L is a uniformly elliptic formally self-adjoint second order differential operator defined on $\Omega.$ Through the viewpoint of functional analysis we define parabolic and hyperbolic operators and show that it is possible to use the detailed analysis of the elliptic system: (UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\eqalign{Lu + f(u)&= g(x)\quad {\rm in}\ \Omega\cr u&=0\quad {\rm on}\ \partial\Omega\cr}$$(TABLE/EQUATION ENDS)to study the time-dependent situation. It turns out that bifurcations from equilibria for periodic solutions of nonlinear reaction diffusion equations and telegraph equations occur precisely at singular points of elliptic operators coming from the elliptic parts of these nonlinear equations. Moreover the fold or cusp type bifurcations occurring for the elliptic problems go to the same fold or cusp bifurcations for the periodic problems. As a consequence of this method we can count exactly how many periodic solutions bifurcate from equilibrium. At the end of this dissertation we give an example of bifurcations of periodic solutions from nonequilibria.
Subject Area
Mathematics
Recommended Citation
Sun, Guozhang, "The new bifurcation method for periodic solutions of time-dependent partial differential equations" (1992). Doctoral Dissertations Available from Proquest. AAI9305904.
https://scholarworks.umass.edu/dissertations/AAI9305904