Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

These investigations and several others that have beenpublishedwithin recentyears compel us us to hold the view that the ratio of the vital capacity to the body length, trunk length, chest circumference,surfacearea or weight or any combination of thesemeasurements, is too variable to admit of any workable standardor normal value. On the other hand the vital capacity of each individual, after he had becomeaccustomedto the use of the spirometer,will be found to be subjectto but small variations as long as good health is maintained. Thereseems to beevidenceto show that a reductionin the vital capacityis ofen the first sign of a progressivedamageto the respiratorytissue.

The cancer cell

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions

We prove the global existence, along with some basic boundedness properties, of weak solutions to a PDE-ODE system modeling the multiscale invasion of tumor cells through the surrounding tissue matrix. The model has been proposed in [G. Meral, C. Stinner, and C. Surulescu, On a Multiscale Model Involving Cell Contractivity and its Effects on Tumor Invasion, preprint, TU Kaiserslautern, Kaiserslautern, Germany, 2013] and accounts on the macroscopic level for the evolution of cell and tissue densities, along with the concentration of a chemoattractant, while on the subcellular level it involves the binding of integrins to soluble and insoluble components of the peritumoral region. The connection between the two scales is realized with the aid of a contractivity function characterizing the ability of the tumor cells to adapt their motility behavior to their subcellular dynamics. The resulting system, consisting of three partial and three ordinary differential equations including a temporal delay, in particul...

https://kluedo.ub.uni-kl.de/files/3621/ssw15.pdf

Global Weak Solutions in a PDE-ODE System Modeling Multiscale Cancer Cell Invasion

New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models

We consider a mathematical model for the spatio-temporal evolution of two biological species in a competitive situation. Besides diffusing, both species move toward higher concentrations of a chemical substance which is produced by themselves. The resulting system consists of two parabolic equations with Lotka–Volterra-type kinetic terms and chemotactic cross-diffusion, along with an elliptic equation describing the behavior of the chemical. We study the question in how far the phenomenon of competitive exclusion occurs in such a context. We identify parameter regimes for which indeed one of the species dies out asymptotically, whereas the other reaches its carrying capacity in the large time limit.

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Competitive exclusion in a two-species chemotaxis model.

We study nonnegative radially symmetric solutions of the chemotaxis system 
(⋆){ut=Δu−χ∇⋅(uv∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,
in a ball Ω⊂Rn, n≥2, with parameter χ>0 and radially symmetric initial data u0∈C0(Ω) and v0∈W1,∞(Ω) satisfying u0≥0 and v0>0 in Ω.

A generalized solution concept is introduced for the Neumann problem associated with (⋆), and within this concept global-in-time solutions are shown to exist regardless of the size of χ>0. This extends previous results which assert global existence of some weak solutions in the case χ<n+23n−4.

Moreover, the obtained solutions are shown to satisfy ∫Ωuθ(x,t)dx 0 and each θ<nn−2⋅min{1,1χ2}. In particular, in the two-dimensional setting this rules out the possibility of what we call a finite-time collapse into persistent Dirac-type singularities.

The construction of such solutions is based on the combination of some weighted integral estimates with a refined Hardy–Sobolev inequality.

Christian Stinner

Papers

Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions

Global Weak Solutions in a PDE-ODE System Modeling Multiscale Cancer Cell Invasion

New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models

Competitive exclusion in a two-species chemotaxis model.

Global weak solutions in a chemotaxis system with large singular sensitivity