Pointwise Bounds for Strongly Coupled Time Dependent Systems of Reaction-Diffusion Equations

1984 ◽  
Vol 15 (2) ◽  
pp. 350-356 ◽  
Author(s):  
Chris Cosner
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Strongly Coupled

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

Comparison principles for strongly coupled reaction diffusion equations in unbounded domains

1988 ◽  
Vol 110 (3-4) ◽  
pp. 311-319 ◽  
Author(s):  
E. Tuma
Keyword(s):  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nerve Fibres ◽  
Strongly Coupled ◽  
Comparison Principles ◽  
Nagumo Equations ◽  
And Diffusion ◽  
Coupled Reaction

SynopsisComparison principles for systems of reaction–diffusion equations in unbounded domains and coupledvia both reaction and diffusion terms are considered. Applications are made to the FitzHugh–Nagumo equations and models of coupled nerve fibres.

scholarly journals Projected Finite Elements for Systems of Reaction-Diffusion Equations on Closed Evolving Spheroidal Surfaces

2017 ◽  
Vol 21 (3) ◽  
pp. 718-747 ◽  
Author(s):  
Necibe Tuncer ◽  
Anotida Madzvamuse
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Projection Operator ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Surface Evolution ◽  
Element Discretization ◽  
Element Method

AbstractThe focus of this article is to present the projected finite element method for solving systems of reaction-diffusion equations on evolving closed spheroidal surfaces with applications to pattern formation. The advantages of the projected finite element method are that it is easy to implement and that it provides a conforming finite element discretization which is “logically” rectangular. Furthermore, the surface is not approximated but described exactly through the projection. The surface evolution law is incorporated into the projection operator resulting in a time-dependent operator. The time-dependent projection operator is composed of the radial projection with a Lipschitz continuous mapping. The projection operator is used to generate the surface mesh whose connectivity remains constant during the evolution of the surface. To illustrate the methodology several numerical experiments are exhibited for different surface evolution laws such as uniform isotropic (linear, logistic and exponential), anisotropic, and concentration-driven. This numerical methodology allows us to study new reaction-kinetics that only give rise to patterning in the presence of surface evolution such as theactivator-activatorandshort-range inhibition; long-range activation.

Quantitative maximum principles and strongly coupled gradient-like reaction-diffusion systems

1983 ◽  
Vol 94 (3-4) ◽  
pp. 265-286 ◽  
Author(s):  
Nicholas D. Alikakos
Keyword(s):  
Reaction Diffusion ◽  
Parabolic Systems ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Diffusion Matrix ◽  
Strongly Coupled ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
All Solutions

SynopsisIn §§1 and 2, we consider mainly a system of reaction-diffusion equations with general diffusion matrix and we establish the stabilization of all solutions at t →∞. The interest of this problem derives from two separate facts. First, the sets that are useful for localizing the asymptotics cease to be invariant as soon as the diffusion matrix is not a multiple of the identity. Second, the set of equilibria is connected. In §3, we establish uniform L§ bounds for the solutions of a class of parabolic systems. The unifying feature in the problems considered is the lack of any conventional maximum principles.

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