Traveling waves for a Fisher-type reaction-diffusion equation with a flux in divergence form Margarita Arias, Juan Campos Mathematical Models and Methods in Applied Sciences, 2023 Analysis of the speed of propagation in parabolic operators is frequently carried out considering the minimal speed at which its traveling waves (TWs) move. This value depends on the solution concept being considered. We analyze an extensive class of Fisher-type reaction–diffusion equations with flows in divergence form. We work with regular flows, which may not meet the standard elliptical conditions, but without other types of singularities. We show that the range of speeds at which classic TWs move is an interval unbounded to the right. Contrary to classic examples, the infimum may not be reached. When the flow is elliptic or over-elliptic, the minimum speed of propagation is achieved. The classic TW speed threshold is complemented by another value by analyzing an extension of the first-order boundary value problem to which the classic case is reduced. This singular minimum speed can be justified as a viscous limit of classic minimal speeds in elliptic or over-elliptic flows. We construct a singular profile for each speed between the minimum singular speed and the speeds at which classic TWs move. Under additional assumptions, the constructed profile can be justified as that of a TW of the starting equation in the framework of bounded variation functions. We also show that saturated fronts verifying the Rankine–Hugoniot condition can appear for strictly lower speeds even in the framework of bounded variation functions.
Cross-diffusion and traveling waves in porous-media flux-saturated Keller-Segel models Margarita Arias, Juan Campos, Juan Soler Mathematical Models and Methods in Applied Sciences, 2018 This paper deals with the analysis of qualitative properties involved in the dynamics of Keller–Segel type systems in which the diffusion mechanisms of the cells are driven by porous-media flux-saturated phenomena. We study the regularization inside the support of a solution with jump discontinuity at the boundary of the support. We analyze the behavior of the size of the support and blow-up of the solution, and the possible convergence in finite time toward a Dirac mass in terms of the three constants of the system: the mass, the flux-saturated characteristic speed, and the chemoattractant sensitivity constant. These constants of motion also characterize the dynamics of regular and singular traveling waves.
Fastness and continuous dependence in front propagation in Fisher-KPP equations Margarita Arias, Juan Campos, Cristina Marcelli, and Discrete and Continuous Dynamical Systems Series B, 2009 We investigate the continuous dependence of the minimal speed of propagation and the profile of the corresponding travelling wave solution of Fisher-type reaction-diffusion equations $\\vartheta_t = (D(\\vartheta)\\vartheta_x)_x + f(\\vartheta)$ with respect to both the reaction term $f$ and the diffusivity $D$. We also introduce and discuss the concept of fast heteroclinic in this context, which allows to interpret the appearance of sharp heteroclinic in the case of degenerate diffusivity ($D(0)=0)$.
The functional Fučik spectrum has empty interior M. Arias, J. Campos, M. Cuesta, J.-P. Gossez Royal Society of Edinburgh Proceedings A, 2003 We define a functional version of the Fučik spectrum for the Laplacian and we prove that this functional spectrum has empty interior.
