Gasless combustion fronts with heat loss (Q2850692)

The paper describes some qualitative properties of a model accounting for gasless combustion. The authors consider the coupled evolution system NEWLINE\[NEWLINE\begin{aligned} \partial _{t}u_{1}&=\partial _{xx}u_{1}+u_{2}\rho (u_{1}-\overline{u} _{1})-\delta u_{1},\\ \partial _{t}u_{2}&=\kappa \partial _{xx}u_{2}-\beta u_{2}\rho (u_{1}-\overline{u}_{1}),\end{aligned}NEWLINE\]NEWLINE where \(\beta >0\), \(\delta \geq 0\), \( \kappa \geq 0\) and \(\rho =e^{-1/u}\) if \(u>0\) and \(\rho =0\) if \(u=0\). Here, \( u_{1}\) is the temperature, \(u_{2}\) is the reactant concentration and \( \overline{u}_{1}\) is the temperature below which the reaction does not occur. The term \(\delta u_{1}\) represents a heat loss. The authors focus on existence, uniqueness and stability properties of traveling combustion fronts, that is of traveling waves, for this system. The paper starts with a review of the literature on the subject, the authors distinguishing between the cases \(\delta =0\) and \(\delta >0\) and here further between \(\kappa =0\) and \(\kappa >0\). The case of traveling waves with speed of order one is considered, introducing the new variable \(\xi =x-\sigma t\) for a finite velocity \(\sigma \). They end with the coupled system of ODEs NEWLINE\[NEWLINE\begin{aligned} 0&=\partial _{\xi \xi }u_{1}+\sigma \partial _{\xi }u_{1}+u_{2}\rho (u_{1}- \overline{u}_{1})-\delta u_{1},\\ 0&=\kappa \partial _{\xi \xi }u_{2}+\sigma \partial _{\xi }u_{2}-\beta u_{2}\rho (u_{1}-\overline{u}_{1}).\end{aligned}NEWLINE\]NEWLINE They start with the case \(\kappa =0=\delta \) and prove the existence of a unique finite velocity \(\sigma =\sigma _{0}(\beta ,\overline{u}_{1})\) depending smoothly on its arguments such that the above gasless combustion model has a traveling wave of speed \(\sigma \). The authors establish a stability result for this traveling wave. Similar results are then proved for the case \( \kappa >0\). Introducing the solution \((u_{1\ast },u_{2\ast })\) of the system of ODEs, the authors build an equilibrium solution NEWLINE\[NEWLINEY_{\ast }(\xi )=(u_{1\ast },z_{2\ast })NEWLINE\]NEWLINE of the modified problem NEWLINE\[NEWLINE\begin{aligned}\partial _{t}u_{1}&=\partial _{\xi \xi }u_{1}+\sigma \partial _{\xi }u_{1}+(u_{2}^{\ast }+z_{2})\rho (u_{1}-\overline{u}_{1})-\delta u_{1}, \\ \partial _{t}z_{2}&=\kappa \partial _{\xi \xi }z_{2}+\sigma \partial _{\xi }z_{\acute{e}}-\beta (u_{2}^{\ast }+z_{2})\rho (u_{1}-\overline{u}_{1}),\end{aligned}NEWLINE\]NEWLINE where \(u_{2}^{\ast }\) is an initial unburned reactant concentration. This system may be written in a vectorial way as NEWLINE\[NEWLINEY(t)=(u_{1},z_{2})^{T}=DY_{\xi \xi }+R(Y),NEWLINE\]NEWLINE where \(D\) is the associated diagonal matrix and \(R\) is the remaining term. The linearization of this problem around \(Y_{\ast }\) leads to the definition of a linear operator \(L\) whose spectrum is indicated in some weighted space. The main result of the paper proves a stability property of this last system, assuming that the Evans function for the traveling wave \(Y_{\ast }(\xi )\) has no zeros in the half-plane \(\operatorname{Re}\lambda >0 \) other than a simple zero at the origin. In the last part of their paper, the authors present some numerical computations in the case \(\kappa =0 \) which prove that the above criterion is satisfied for the Evans function.