Bifurcation Behavior Of Nonconstant Steady State Solutions Of General Brusselator Models

In this paper,bifurcation behavior of nonconstant steady state solutions for general Brusselator model are studied by a priori estimate,Rabinowitz bifurcation theorem and Leray-Schauder degree theory.The main works are as follows:1.We consider the steady state problem corresponding to the general Brusse-lator model where a>0 are fixed parameters with d2>d1,b>0 is a bifurcation param-eter.We obtain the local and global bifurcation of nonnegative nonconstant steady state solutions under conditions that h ?C([0,?),[0,?)is a strictly increasing function and h(s)/s2 is non-increasing in(0,?).When h(u)?u,the system degenerated into the problem studied by Ma and Hu in 2014.The results in this part generalized their first work and corrected their second theorem.2.We consider the steady state problem corresponding to the general coupled two-cell Brusselator model where d1,d2,a,c>0 are fixed parameters with d2>d1,b>0 is a bifurcation parameter.The model describes the interaction between the two cells as well as the internal reactions,u and v are two substances in the same cell,w and z are two substances in the other cell,and c(w-u)is the coupling term.We obtain the local and global bifurcation of nonnegative nonconstant solutions under conditions that f?([0,?),[0,?)),f'(a)?(0,?)and is non-decreasing in(0,?).The main difficulties to be overcome in this part are the priori estimation of solutions and the calculation of matrix eigenvalues caused by the increase of the number of equations and the disturbance of coupling terms.3.We consider the steady state problem corresponding to the general coupled two-cell Brusselator model where d1,d2,a,b,c>0 are fixed parameters with d2>d1,?>0 is a bifurcation pa-rameter.We get the nonexistence of the nonconstant solutions when ?>0 very smal-1.When the parameters a.b,c,d1,d2 satisfy a certain relation,we obtain the structure of the nonconstant solutions set under conditions that f?C1G[0,?),[0,?)),f'(a)?(0,?)and is non-decreasing in(0,?).When f(u)=u2,the system degenerates into the problem studied by Zhou and Mu in 2010.