## 2 Materials and Methods
Therefore, for sufficiently continuous initial data, i.e., in the domain, the evolving solutions \(N(x,t)\) and \(\Theta(x,t)\) are continuous and smooth (Chapter 7 [21]) in virtue of the regular parabolic spatial operator that adopts the Laplacian form as it will be shown. Then:  
\[\frac{\partial}{\partial t}\int_{\Sigma}N\,dx=-\int_{\partial\Sigma}\Psi_{N} \cdot\pi\,dA+\int_{\Sigma}G_{N}(N,\Theta,x,t)\,dx, \tag{4}\]  
\[\frac{\partial}{\partial t}\int_{\Sigma}\Theta\,dx=-\int_{\partial\Sigma}\Psi_ {\Theta}\cdot\pi\,dA+\int_{\Sigma}G_{\Theta}(N,\Theta,x,t)\,dx. \tag{5}\]  
The Fick law [21] enables us to account for an expression to the vector fluxes:  
\[\Psi_{N}=-\rho\
abla N-aN,\;\;\Psi_{\Theta}=-\sigma\
abla\Theta-a\Theta. \tag{6}\]  
The diffusion between the two interacting species requires that \(\rho=\sigma\). Note that the vector \(a\in\mathds{R}^{3}\) introduces the advection term to account for the forced convection within the fuel tank. Hence:  
\[\frac{\partial}{\partial t}\int_{\Sigma}N\,dx=-\int_{\partial\Sigma}(-\rho \
abla N-aN)\cdot\pi\,dA+\int_{\Sigma}G_{N}(N,\Theta,x,t)\,dx \tag{7}\]  
\[\frac{\partial}{\partial t}\int_{\Sigma}\Theta\,dx=-\int_{\partial\Sigma}(- \sigma\
abla\Theta-a\Theta)\cdot\pi\,dA+\int_{\Sigma}G_{\Theta}(N,\Theta,x,t) \,dx. \tag{8}\]  
Note that:  
\[\int_{\partial\Sigma}(-\rho\
abla N-aN)\cdot\pi dA=\int_{\Sigma}\
abla\cdot(- \rho\
abla N-aN)dx=\int_{\Omega}(-\rho\
abla\cdot(\
abla N)-a\cdot\
abla N)dx, \tag{9}\]  
\[\int_{\partial\Sigma}(-\sigma\
abla\Theta-a\Theta)\cdot\pi dA=\int_{\Sigma} \
abla\cdot(-\sigma\
abla\Theta-a\Theta)dx=\int_{\Sigma}(-\sigma\
abla\cdot( \
abla\Theta)-a\cdot\
abla\Theta)dx. \tag{10}\]  
So that:  
\[\frac{\partial}{\partial t}\int_{\Sigma}N\,dx=-\int_{\Sigma}(-\rho\Delta N-a \cdot\
abla N)\,dx+\int_{\Sigma}G_{N}(N,\Theta,x,t)\,dx, \tag{11}\]  
\[\Theta_{t}=-N^{m}. \tag{3}\]0  
The initial concentrations of nitrogen and oxygen are mathematically given by constant homogeneous distributions, i.e., \(N_{0}(x)=N_{0}>0\), \(\Theta_{0}(x)=\Theta_{0}>0\):  
\[\Theta_{t}=-N^{m}. \tag{3}\]1  
Considering the above information, the following problem \(P\) is formulated:  
\[ \begin{array}{c}N_{t}=\rho\Delta N+a\cdot\
abla N+\Theta^{n},\\ \Theta_{t}=\sigma\Delta\Theta+a\cdot\
abla\Theta-N^{m},\\ 0<n,\,m<1\\ N_{0}(x)>0,\Theta_{0}(x)>0\in L^{\infty}(\mathds{R}^{3})\cap L^{1}_{loc}( \mathds{R}^{3}).\end{array}  \tag{14}\]  
The operator's parabolic regularity ensures the monotone evolution of each of the involved solutions (referring to Chapter 7 [21]). The decreasing and increasing condition of the solutions are, then, given by the reaction (increasing) and absorption terms (decreasing). As a consequence and preliminary, the nitrogen concentration \(N\) increases while the oxygen concentration \(\Theta\) decreases. Then, and upon evolution, the solutions will satisfy \(N(x,t),\ \Theta(x,t)\in L^{1}(\mathrm{R}^{3})\), which means that the total concentration of each substance does not blow up in the domain due to the non-linearity proposed in the reaction and absorption terms. The problem \(P\) was introduced as a baseline model to account for the different terms involved in the gases dynamics. Now, a new problem is defined accordingly to account for the asymptotic behaviour of each of the substances. As it will be shown in Section 3.3, the fuel tank inerting process tends towards stationary values for the oxygen and nitrogen concentrations. This fact leads to consider travelling waves (TW) solutions connecting two stationary concentrations, both given by the initial and the asymptotic values. Mathematically, such solutions can be modelled by a Fisher-KPP reaction type. Consequently, the new problem in the TW domain \(P_{T}\) is:  
\[ \begin{array}{c}N_{t}=\rho\Delta N+a\cdot\
abla N-\Theta^{n}(N-r),\\ \Theta_{t}=\sigma\Delta\Theta+a\cdot\
abla\Theta-N^{m}\Theta,\\ 0<n,m<1,\\ N_{0}(x),\Theta_{0}(x)\in L^{\infty}(\mathrm{R}^{3})\cap L^{1}_{loc}( \mathrm{R}^{3}),\end{array}  \tag{15}\]  
where \(r\geq\max\{N_{0}(x)\}\) is the asymptotic solution to \(N\).