Journal ArticleDOI
Global Existence in Reaction-Diffusion Systems with Control of Mass: a Survey
TLDR
In this article, the authors describe the state of the art on the question of global existence of solutions to reaction-diffusion systems for which two main properties hold: on one hand, the positivity of the solutions is preserved for all time; on the other hand the total mass of the components is uniformly controlled in time.Abstract:
The goal of this paper is to describe the state of the art on the question of global existence of solutions to reaction-diffusion systems for which two main properties hold: on one hand, the positivity of the solutions is preserved for all time; on the other hand, the total mass of the components is uniformly controlled in time. This uniform control on the mass (or – in mathematical terms- on the L1-norm of the solution) suggests that no blow up should occur in finite time. It turns out that the situation is not so simple. This explains why so many partial results in different directions are found in the literature on this topic, and why also the general question of global existence is still open, while lots of systems arise in applications with these two natural properties. We recall here the main positive and negative results on global existence, together with many references, a description of the still open problems and a few new results as well.read more
Citations
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Book
Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States
Pavol Quittner,Philippe Souplet +1 more
TL;DR: In this article, the authors propose a model for solving the model elliptic problems and model parabolic problems. But their model is based on Equations with Gradient Terms (EGS).
Book ChapterDOI
Parabolic Equations in Biology
TL;DR: This chapter contains a general presentation of parabolic partial differential equations that are used in biology: Lotka-Volterra systems and chemical or enzymatic reactions, or in a mathematical classification, semilinear equations.
Journal ArticleDOI
Improved Duality Estimates and Applications to Reaction-Diffusion Equations
TL;DR: For general systems in any space dimension, this article obtained smooth solutions of reaction-diffusion systems coming out of reversible chemistry under the assumption that the diffusion coefficients are sufficiently close one to another.
Journal ArticleDOI
Global Existence of Renormalized Solutions to Entropy-Dissipating Reaction–Diffusion Systems
TL;DR: In this article, the authors introduce the notion of a renormalized solution for reaction-diffusion systems with entropy-dissipating reactions and establish the global existence of such solutions.
Journal ArticleDOI
Global Existence Analysis of Cross-Diffusion Population Systems for Multiple Species
TL;DR: The existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of competing population species is proved in this paper, based on a refined entropy method and a new approximation scheme.
References
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Book
Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Journal ArticleDOI
The Chemical Basis of Morphogenesis
TL;DR: In this article, it is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.
Book
Geometric Theory of Semilinear Parabolic Equations
TL;DR: The neighborhood of an invariant manifold near an equilibrium point is a neighborhood of nonlinear parabolic equations in physical, biological and engineering problems as mentioned in this paper, where the neighborhood of a periodic solution is defined by the invariance of the manifold.
Journal ArticleDOI
The Chemical Basis of Morphogenesis
TL;DR: A possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism is discussed, suggesting that certain well-known physical laws are sufficient to account for many of the facts.
MonographDOI
Linear and Quasi-linear Equations of Parabolic Type
TL;DR: In this article, the authors introduce a system of linear and quasi-linear equations with principal part in divergence (PCI) in the form of systems of linear, quasilinear and general systems.