An upper bound for the bulk burning rate for systems (Q2757161)

The authors consider a system of reaction-diffusion equations with passive advection term and Lewis number \(Le\): NEWLINE\[NEWLINE \begin{cases} T_t+u\cdot \nabla T=k\Delta T+{v_0^2\over k}g(T)n,\\ n_t+u\cdot \nabla n={k\over Le}\Delta n- {v_0^2 \over k}g(T)n.\end{cases} NEWLINE\]NEWLINE The velocity \(u\) is passive and presumed to be given. The nonlinearity \(g(T)\) is assumed to be of the KPP-type: NEWLINE\[NEWLINE g(0)=0,\quad g'(0)\not= 0,\quad g(T)\leq g'(0)T\quad \text{ for}\quad T>0. NEWLINE\]NEWLINE It is proved a general upper bound on the reaction rate in such systems in terms of the reaction rate for a single reaction-diffusion equation, showing that the long-time average of the reaction rate with \(Le\not= 1\) does not exceed the \(Le=1\) case. The upper estimates derived for \(Le=1\) apply to systems with both front-like and compact initial data.