Quenching and propagation of combustion fronts in porous media (Q879651)

The author is concerned with Sivashinsky's model for subsonic detonation: \[ \begin{cases} \gamma T_y-(\gamma-1)P_t=\varepsilon\Delta T+Y\Omega(T)\\ P_t-T_t=\Delta P\\ Y_t=\varepsilon Le^{-1}\Delta Y-Y\Omega(T) \end{cases}\tag{S} \] where \(T, P, Y\) refer to the scaled temperature, pressure and concentration reactant respectively. \(\gamma>1\) is the specific heat ratio, \(Le\) is the Lewis number and \(\varepsilon\) is a ratio of pressure and molecular diffusivities. The reaction rate \(\Omega\) obeys the Arrhenius Law. The system (S) is subject to the initial conditions: \[ T(0,x)=T_0(x),\quad Y(0,x)=1,\quad P(0,x)=0. \] The author first makes a linear transform \((T,P)\rightarrow(R,S)\) so that the system (S) is rewritten as \[ \begin{cases} R_t=\delta\Delta R+Y\Omega(T)\\ S_t=\Delta S+\frac{1}{1-\delta}Y\Omega(T)\\ Y_t=\mu\Delta Y-Y\Omega(T)\\ T(x,t)=\lambda S(t,x)+ (1-\lambda)-\frac{\lambda\delta}{1-\delta}R(t,x) \end{cases} \] where \(\delta\) and \(\mu\) are new constants. The latter system is then formulated as integral equations for the unknowns \(R\) and \(S\) by means of the heat kernel in \(\mathbb R^d.\) This allows to get estimates of \(S, Y\) and then to prove that: (1) Small \(T\) leads to quenching (decay of \(T\) and \(P\) to zero as t goes to infinity). (2) \(T\) large enough imply diffusive propagation of disturbances with finite speed.