Bifurcation For A Free Boundary Problem Modeling A Protocell

In this dissertation we study some free boundary problems arising from the field of cell growth modelling. As we all known, any organism maintains its identity as a finite and delimited system by pursuing a metabolism to such a degree that typical forms and sizes are developed and possibly stabilized. The aim of the work reported below is to study this ability in a free boundary problem for physicochemicalmodel of a protocell. The model of a self-maintaining unity or a protocell is based on the reaction and diffusion process, and a mechanism of self-control of the boundary. The special case of radially symmetric cells was studied in earlier works. The present paper is concerned with the existence of symmetry-breaking stationary solutions, i.e., with solutions which are not radially symmetric. It is proved^[12,13], for any positive radius R, there exists symmetry-breaking solution in two-dimensional space. The purpose of the present paper is two folds: firstly, we want to extend the bifurcation results of ^[12,13]into the more realistic three dimensional case; secondly, we shall use a much more simplified and systematic method to establish the existence of symmetry-breaking bifurcation branches of stationary solutions by applying the Crandall-Rabinowitz bifurcation theorem.This paper is organized as follows: In section 1, we provide some background material and collect various results on the Bessel functions that are required in the subsequent sections. In section 2, we construct the bifurcation problem of protoell modelling. In section 3-5, we shall see that F maps the space C^m+α(∑)×R into the space C^m+α-1(∑). In sections 4 we shall formally compute the Frechet derivatives of F. Since F is smooth, the formal derivation will be rigorously justified in section 3.In the end, we proved there exist branches of non-radial stationary solutions bifurcating from radially symmetric solutions in the more realistic three space dimensional case; indeed, for any mode n, n≥2, there exists a unique bifurcation branch whose free boundary has the form r = R +εY_n,0(θ) + O(ε²) (n is even) for small |ε|, where Y_n,0 is the spherical harmonic of mode (n, 0).