A non-local bistable reaction-diffusion equation with a gap
Henri Berestycki Nancy Rodríguez
Discrete & Continuous Dynamical Systems - A 2017, 37(2): 685-723 doi: 10.3934/dcds.2017029

Non-local reaction-diffusion equations arise naturally to account for diffusions involving jumps rather than local diffusions related to Brownian motion. In ecology, long distance dispersal require such frameworks. In this work we study a one-dimensional non-local reaction-diffusion equation with bistable type reaction. The heterogeneity here is due to a gap, some finite region where there is decay. Outside this gap region the equation is a classical homogeneous (space independent) non-local reaction-diffusion equation. This type of problem is motivated by applications in ecology, sociology, and physiology. We first establish the existence of a generalized traveling front that approaches a traveling wave solution as t-∞, propagating in a heterogeneous environment. We then study the problem of obstruction of solutions. In particular, we study the propagation properties of the generalized traveling front with significant use of the work of Bates, Fife, Ren and Wang in [5]. As in the local diffusion case, we prove that obstruction is possible if the gap is sufficiently large. An interesting difference between the local dispersal and the non-local dispersal is that in the latter the obstructing steady states are discontinuous. We characterize these jump discontinuities and discuss the scaling between the range of the dispersal and the critical length of the gap observed numerically. We further explore other differences between the local and the non-local dispersal cases. In this paper, we illustrate these properties by numerical simulations and we state a series of open problems.

keywords: Entire solution gap problem non-local diffusion comparison principle propagation
Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $\mathbb{R}^d$
Jacob Bedrossian Nancy Rodríguez
Discrete & Continuous Dynamical Systems - B 2014, 19(5): 1279-1309 doi: 10.3934/dcdsb.2014.19.1279
Aggregation equations and parabolic-elliptic Patlak-Keller-Segel (PKS) systems for chemotaxis with nonlinear diffusion are popular models for nonlocal aggregation phenomenon and are a source of many interesting mathematical problems in nonlinear PDEs. The purpose of this work is to give a more complete study of local, subcritical and small-data critical/supercritical theory in $\mathbb{R}^d$, $d \geq 2$. Some existing results can be found in the literature; however, one of the most important cases in biological applications, that is the $\mathbb{R}^2$ case, had not been studied. In this paper, we treat two related systems, which are different generalizations of the classical parabolic-elliptic PKS model. In the first class, nonlocal aggregation is modeled by convolution with a general interaction potential, studied in this generality in our previous work [6]. For this class of models we also present several large data global existence results for critical problems. The second class is a variety of PKS models with spatially inhomogeneous diffusion and decay rate of the chemo-attractant, which is potentially relevant to biological applications and raises interesting mathematical questions.
keywords: inhomogeneous Patlak-Keller-Segel Global well-posedness degenerate diffusion aggregation equation parabilic equations.

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