Stability analysis for combustion fronts traveling in hydraulically resistant porous media

We study front solutions of a system that models combustion in highly hydraulically resistant porous media. The spectral stability of the fronts is tackled by a combination of energy estimates and numerical Evans function computations. Our results suggest that there is a parameter regime for which there are no unstable eigenvalues. We use recent works about partially parabolic systems to prove that in the absence of unstable eigenvalues the fronts are convectively stable.Comment: 21 pages, 4 figure

Fisher-KPP dynamics in diffusive Rosenzweig-MacArthur and Holling-Tanner models

We prove the existence of traveling fronts in diffusive Rosenzweig-MacArthur and Holling-Tanner population models and investigate their relation with fronts in a scalar Fisher-KPP equation. More precisely, we prove the existence of fronts in a Rosenzweig-MacArthur predator-prey model in two situations: when the prey diffuses at the rate much smaller than that of the predator and when both the predator and the prey diffuse very slowly. Both situations are captured as singular perturbations of the associated limiting systems. In the first situation we demonstrate clear relations of the fronts with the fronts in a scalar Fisher-KPP equation. Indeed, we show that the underlying dynamical system in a singular limit is reduced to a scalar Fisher-KPP equation and the fronts supported by the full system are small perturbations of the Fisher-KPP fronts. We obtain a similar result for a diffusive Holling-Tanner population model. In the second situation for the Rosenzweig-MacArthur model we prove the existence of the fronts but without observing a direct relation with Fisher-KPP equation. The analysis suggests that, in a variety of reaction-diffusion systems that rise in population modeling, parameter regimes may be found when the dynamics of the system is inherited from the scalar Fisher-KPP equation

On the stability of high Lewis number combustion fronts

We consider wavefronts that arise in a mathematical model for high Lewis number combustion processes. An e�cient method for the proof of the existence and uniqueness of combustion fronts is provided by geometric singular perturbation theory. The fronts supported by the model with very large Lewis numbers are small perturbations of the front supported by the model with in nite Lewis number. The question of stability for the fronts is more complicated. Besides discrete spectrum, the system possesses essential spectrum up to the imaginary axis. We show how a geometric approach which involves construction of the Stability Index Bundles can be used to relate the spectral stability of wavefronts with high Lewis number to the spectral stability of the front in the case of in nite Lewis number

Nonlinear stability of high Lewis number combustion fronts

In this paper a mathematical model is considered that describes combustion processes characterized by a very high Lewis number. The model is known to support a wavefront that asymptotically connects the completely burned and the unburned states, and that is unique up to translation. The stability of the front has not yet been investigated beyond the spectral level. The essential spectrum of the linearization of the system about the front touches the imaginary axis, therefore, even in a parameter regime that guarantees absence of the unstable discrete spectrum, spectral information is not definitive. There exists an exponentially weighted norm that stabilizes the front on the linear level. The nonlinear stability in that exponentially weighted norm cannot be simply inferred from the spectral stability because the nonlinearity is not smooth in that norm. We use the interplay of the norms with and without exponential weight to overcome this issue, and show that the front in the co-moving frame is nonlinearly stable in the exponentially weighted norm with respect to a special class of perturbations

Nonlinear Convective Instability of Turing‐Unstable Fronts near Onset: A Case Study

Fronts are travelling waves in spatially extended systems that connect two different spatially homogeneous rest states. If the rest state behind the front undergoes a supercritical Turing instability, then the front will also destabilize. On the linear level, however, the front will be only convectively unstable since perturbations will be pushed away from the front as it propagates. In other words, perturbations may grow but they can do so only behind the front. The goal of this paper is to prove for a specific model system that this behaviour carries over to the full nonlinear system