Travelling wave solutions in a model for filtration combustion
The existence and uniqueness of travelling wave solutions of the evolution system \[ \gamma\widehat\theta_t-(\gamma-1)\widehat\Pi_t= \Omega(\widehat\varphi,\widehat\theta),\quad \widehat\varphi_t=-\Omega(\widehat\varphi,\widehat\theta), \quad \widehat\Pi_t- \widehat\theta_t= \widehat\Pi_{xx}, \] for \(x\in\mathbb{R}\) and \(t>0\), are proved. This system arises in a combustion model proposed by [\textit{I.~Brailovsky, M.~Frankel} and \textit{G. I.~Sivashinsky} [Combust. Theory Model 4, 47--60 (2000; Zbl 0966.76098)]. The variables are here the temperature \(\widehat\theta(x,t)\), the pressure \(\widehat\Pi(x,t)\) and the concentration of the deficient reactant \(\widehat\varphi(x,t)\). The term \(\Omega(\widehat\varphi,\widehat\theta)\) is the nonlinear reaction term and the constant \(\gamma>1\) only depends on the reacting mixture. The author is interested in travelling wave solutions to this system of the form \(\widehat\theta(x,t)=\theta(x+ct)\), \(\widehat\Pi(x,t)=\Pi(x+ct)\), \(\widehat\varphi(x,t)=\varphi(x+ct)\), where the number \(c\) is proportional to the propagation speed of the combustion front. An estimate of the wave speed is established, too.
scientific article; zbMATH DE number 2102592