Invariant regions for quasilinear reaction-diffusion systems and applications to a two population model

1996 ◽  
Vol 3 (4) ◽  
pp. 421-444 ◽  
Author(s):  
Konrad Horst Wilhelm K�fner
Keyword(s):  
Population Model ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Invariant Regions

Contracting sets and dissipation

1995 ◽  
Vol 125 (6) ◽  
pp. 1305-1329 ◽  
Author(s):  
Alexandre N. Carvalho
Keyword(s):  
Partial Differential Equations ◽  
Fractional Power ◽  
Reaction Diffusion ◽  
Global Attractors ◽  
Parabolic Partial Differential Equations ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Variation Of Constants Formula ◽  
Variation Of Constants ◽  
Invariant Regions

In this work we study reaction–diffusion systems in fractional power spaces Xα which are embedded in L∞. We prove that the solution operators T(t) to these problems are globally defined, point dissipative, locally bounded and compact. That ensures the existence of global attractors. We also find a set containing the range of every function in the attractor, providing good estimates on asymptotic concentrations. This is done under very few hypotheses on the reaction term. These hypotheses are natural and easy to verify in many applications. The tools employed are the theory of invariant regions for systems of parabolic partial differential equations, the notion of contracting sets and the variation of constants formula. Several examples are considered to emphasise the applicability of these techniques.

Weakly invariant regions for reaction—diffusion systems and applications

2000 ◽  
Vol 130 (5) ◽  
pp. 1165-1180 ◽  
Author(s):  
Si Ning Zheng
Keyword(s):  
Parabolic Equations ◽  
Existence Of Solutions ◽  
Reaction Diffusion ◽  
Chemical Processes ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Systems ◽  
Global Existence Of Solutions ◽  
Diffusion Systems ◽  
Invariant Regions ◽  
Semilinear Parabolic

The important theory of invariant regions in reaction-diffusion equations has only restricted applications because of its strict requirements on both the reaction terms and the regions. The concept of weakly invariant regions was introduced by us to admit wider reaction-diffusion systems. In this paper we first extend the L∞ estimate technique of semilinear parabolic equations of Rothe to the more general case with convection terms, and then propose more precise criteria for the bounded weakly invariant regions. We illustrate, by three model examples, that they are very convenient for establishing the global existence of solutions for reaction-diffusion systems, especially those from ecology and chemical processes.

scholarly journals Numerical Preservation of Velocity Induced Invariant Regions for Reaction–Diffusion Systems on Evolving Surfaces

2018 ◽  
Vol 77 (2) ◽  
pp. 971-1000 ◽  
Author(s):  
Massimo Frittelli ◽  
Anotida Madzvamuse ◽  
Ivonne Sgura ◽  
Chandrasekhar Venkataraman
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Invariant Regions ◽  
Evolving Surfaces

scholarly journals Invariant Regions and Global Existence of Solutions for Reaction-Diffusion Systems with a Tridiagonal Matrix of Diffusion Coefficients and Nonhomogeneous Boundary Conditions

2007 ◽  
Vol 2007 ◽  
pp. 1-15 ◽  
Author(s):  
Abdelmalek Salem
Keyword(s):  
Global Existence ◽  
Diffusion Coefficients ◽  
Existence Of Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Global Existence Of Solutions ◽  
Nonhomogeneous Boundary ◽  
Diffusion Systems ◽  
Invariant Regions ◽  
Nonhomogeneous Boundary Conditions

The purpose of this paper is the construction of invariant regions in which we establish the global existence of solutions for reaction-diffusion systems (three equations) with a tridiagonal matrix of diffusion coefficients and with nonhomogeneous boundary conditions after the work of Kouachi (2004) on the system of reaction diffusion with a full 2-square matrix. Our techniques are based on invariant regions and Lyapunov functional methods. The nonlinear reaction term has been supposed to be of polynomial growth.

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