Local and global bifurcation of steady states to a general Brusselator model

Abstract

Abstract In this paper, we consider the local and global bifurcation of nonnegative nonconstant solutions of a general Brusselator model {−d1△u=a−(b+1)f(u)+u2v,x∈Ω,−d2△v=bf(u)−u2v,x∈Ω,∂u∂n=∂v∂n=0,x∈∂Ω, $$ \textstyle\begin{cases} -d_{1}\triangle u=a-(b+1)f(u)+u^{2}v, & x\in \varOmega , \\ -d_{2}\triangle v=bf(u)-u^{2}v, & x\in \varOmega , \\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in \partial \varOmega , \end{cases} $$ where d1,d2,a>0 $d_{1},d_{2},a>0$ are fixed parameters with d2>d1 $d_{2}>d_{1}$, b>0 $b>0$ is a bifurcation parameter; f∈C([0,∞),[0,∞)) $f\in C([0,\infty ) ,[0,\infty ))$ is a strictly increasing function and f′(f−1(a))∈(0,∞) $f'(f^{-1}(a))\in (0,\infty )$. Moreover, via the Rabinowitz bifurcation theorem, we obtain the global structure of nonconstant solutions under the condition that f(s)s2 $\frac{f(s)}{s^{2}}$ is nonincreasing in (0,∞) $(0,\infty )$.

Keywords

• Brusselator model
• Nonconstant steady state solutions
• Local bifurcation
• Global bifurcation