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A reaction-diffusion-ODE model for the Neolithic spread of farmers in Europe has been recently proposed in [7]. In this model, farmers are assumed to be divided into two subpopulations according to a mobility rule, namely, into sedentary and migrating farming populations. The conversion between the farming subpopulations depends on the total density of farmers and it is superimposed on the classical Lotka-Volterra competition model, so that it is described by a three-component reaction-diffusion-ODE system. In this article we consider a singular limit problem when the conversion rate tends to infinity and prove under appropriate conditions that solutions of the three component system converge to solutions of a two-component system with a linear diffusion and nonlinear degenerate diffusion. [ABSTRACT FROM AUTHOR]

This paper establishes the spectral stability in exponentially weighted spaces of smooth traveling monotone fronts for reaction diffusion equations of Fisher-KPP type with nonlinear degenerate diffusion coefficient. It is assumed that the former is degenerate, that is, it vanishes at zero, which is one of the equilibrium points of the reaction. A parabolic regularization technique is introduced in order to locate a subset of the compression spectrum of the linearized operator around the wave, whereas the point spectrum is proved to be stable with the use of energy estimates. Detailed asymptotic decay estimates of solutions to spectral equations are required in order to close the energy estimates. It is shown that all fronts traveling with speed above a threshold value are spectrally stable in an appropriately chosen exponentially weighted L 2 -space. [ABSTRACT FROM AUTHOR]

This article deals with the mathematical analysis of the forest kinematic model. The model is described by a system of two ordinary differential equations and one parabolic differential equation with nonlinear degenerate diffusion. Three unknown functions represent the tree densities of young and old age classes and the density of seeds. We study the initial boundary value problem for this system with nonnegative initial functions satisfying suitable conditions. We will show the existence of a weak time-global solution which is uniformly bounded. [ABSTRACT FROM AUTHOR]

Continuum mathematical models for collective cell motion normally involve reaction–diffusion equations, such as the Fisher–KPP equation, with a linear diffusion term to describe cell motility and a logistic term to describe cell proliferation. While the Fisher–KPP equation and its generalisations are commonplace, a significant drawback for this family of models is that they are not able to capture the moving fronts that arise in cell invasion applications such as wound healing and tumour growth. An alternative, less common, approach is to include nonlinear degenerate diffusion in the models, such as in the Porous–Fisher equation, since solutions to the corresponding equations have compact support and therefore explicitly allow for moving fronts. We consider here a hole-closing problem for the Porous–Fisherequation whereby there is initially a simply connected region (the hole) with a nonzero population outside of the hole and a zero population inside. We outline how self-similar solutions (of the second kind) describe both circular and non-circular fronts in the hole-closing limit. Further, we present new experimental and theoretical evidence to support the use of nonlinear degenerate diffusion in models for collective cell motion. Our methodology involves setting up a two-dimensional wound healing assay that has the geometry of a hole-closing problem, with cells initially seeded outside of a hole that closes as cells migrate and proliferate. For a particular class of fibroblast cells, the aspect ratio of an initially rectangular wound increases in time, so the wound becomes longer and thinner as it closes; our theoretical analysis shows that this behaviour is consistent with nonlinear degenerate diffusion but is not able to be captured with commonly used linear diffusion. This work is important because it provides a clear test for degenerate diffusion over linear diffusion in cell lines, whereas standard one-dimensional experiments are unfortunately not capable of distinguishing between the two approaches. • The Porous–Fisher equation has a degenerate diffusion term and a logistic growth term. • It allows for solutions with compact support and therefore can capture moving fronts. • Self-similar solutions of the second kind describe circular hole-closing. • Numerical hole-closing solutions compare with new experimental wound healing assays. • We provide a clear test for degenerate diffusion over linear diffusion for cell migration. [ABSTRACT FROM AUTHOR]

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu-Lσ,μ[φ(u)] = f in RN×(0,T), where Lσ,μ is a general symmetric diffusion operator of Lévy type and φ is merely continuous and non-decreasing. We then use this theory to prove convergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion operators Lσ,μ are the (fractional) Laplacians Δ and -(-Δ)α/2 for α ∈ (0,2), discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal Lévy operators, allows us to give a unified and compact nonlocal theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and convergence of the methods under minimal assumptions, including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in [F. del Teso, J. Endal, and E. R. Jakobsen, C. R. Math. Acad. Sci. Paris, 355 (2017), pp. 1154--1160]. We also present some numerical tests, but extensive testing is deferred to the companion paper [F. del Teso, J. Endal, and E. R. Jakobsen, SIAM J. Numer. Anal., 56 (2018), pp. 3611-3647] along with a more detailed discussion of the numerical methods included in our theory. [ABSTRACT FROM AUTHOR]

We develop a unfied and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate di ffusion equations ∂u-L[φ(u)] = f(x; t) in ℝN×(0; T); where L is a general symmetric Lévy-type diffusion operator. Included are both local and nonlocal problems with, e.g., L = Δ or L = -(-Δ) ff2, ff ϵ (0; 2), and porous medium, fast diffusion, and Stefan-type nonlinearities φ . By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are Lp-stable for p ϵ[1,∞], compact, and convergent in C([0; T];Lploc(ℝN)) for p 2 [1,∞). The first part of this project is given in [F. del Teso, J. Endal, and E. R. Jakobsen, preprint, arXiv:1801.07148v1 [math.NA], 2018] and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of Part I apply and testing the schemes numerically. Our examples include fractional diffusions of different orders and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems. [ABSTRACT FROM AUTHOR]

We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem for the anomalous diffusion equation ∂ t u − L μ [ φ ( u ) ] = 0 . Here L μ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function φ : R → R is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain stability, L 1 -contraction, and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Nagumo equation with nonlinear degenerate diffusion and convection which arises in several problems of population dynamics, chemical reactions and others. A sharp front-type solution with a minimum speed to this model equation is analysed using different methods. One of the methods is to solve the travelling wave equations and compute an exact solution which describes the sharp travelling wavefront. The second method is to solve numerically an initial-moving boundary-value problem for the partial differential equation and obtain an approximation for this sharp front-type solution. [ABSTRACT FROM AUTHOR]