Steady states and dynamics of an autocatalytic chemical reaction model with decay (Q432441)

The dynamics and steady state solutions of an autocatalytic chemical reaction model with decay in the catalyst are considered. After rescaling, the model has the following form: NEWLINE\[NEWLINE\begin{cases} u_t=\Delta u+\lambda(1-u)v^p, & x\in \Omega, \;t>0, \\ v_t=D\Delta v+\lambda (1-u)v^p-kv^q, & x\in \Omega, \;t>0, \\ u(x,t)=0, \;v(x,t)=0, & x\in \partial \Omega, \;t>0, \\ u(x,0)=u^0(x)\leq 1, v(x,0)=v^0(x)\geq 0, & x\in \Omega. \end{cases} NEWLINE\]NEWLINE Here \(1-u\) and \(v\) are the concentrations of the reactant and the autocatalyst respectively, \(\lambda\) is the reaction rate, \(k\) is the decay rate of the autocatalyst, \(p,q>1\) are constants, and \(\Omega\) is a bounded domain in \(\mathbb{R}^n\).NEWLINENEWLINELet \(r=\lambda^{q-1}k^{1-p}D^{p-q}\). The authors show that positive solutions exist globally and all steady state solutions are bounded. In the strong decay case, \(q>p\), if \(r\) is small, all solutions converge to the trivial steady state \((0,0)\), so no pattern formation is possible, while if \(r\) is large, some nontrivial steady state solutions exist. Two methods are used to prove the existence of positive steady state solutions: upper-lower solutions and bifurcation theory.