EXPONENTIALLY DECAYING COMPONENT OF A GLOBAL SOLUTION TO A REACTION–DIFFUSION SYSTEM

1998 ◽  
Vol 08 (05) ◽  
pp. 897-904 ◽  
Author(s):  
HIROKI HOSHINO ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Global Solution ◽  
Large Time ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Spatial Average ◽  
Exponential Rate ◽  
Chemical Substances ◽  
Irreversible Chemical Reaction ◽  
The Difference

A reaction–diffusion system which is related to a simple irreversible chemical reaction between two chemical substances is considered. When a non-negative global solution for the system converges uniformly to zero with polynomial rate as time goes to infinity, large-time approximation of the solution is studied. It is shown that the difference of the solution and its spatial average tends to zero with exponential rate via a global solution for the corresponding system of ordinary differential equations.

scholarly journals A necessary and sufficient condition for global existence for a quasilinear reaction-diffusion system

2005 ◽  
Vol 2005 (11) ◽  
pp. 1809-1818 ◽  
Author(s):  
Alan V. Lair
Keyword(s):  
Global Solution ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Special Case

We show that the reaction-diffusion systemut=Δφ(u)+f(v),vt=Δψ(v)+g(u), with homogeneous Neumann boundary conditions, has a positive global solution onΩ×[0,∞)if and only if∫∞ds/f(F−1(G(s)))=∞(or, equivalently,∫∞ds/g(G−1(F(s)))=∞), whereF(s)=∫0sf(r)drandG(s)=∫0sg(r)dr. The domainΩ⊆ℝN(N≥1)is bounded with smooth boundary. The functionsφ,ψ,f, andgare nondecreasing, nonnegativeC([0,∞))functions satisfyingφ(s)ψ(s)f(s)g(s)>0fors>0andφ(0)=ψ(0)=0. Applied to the special casef(s)=spandg(s)=sq,p>0,q>0, our result proves that the system has a global solution if and only ifpq≤1.

scholarly journals Existence of global solution and nontrivial steady states for a system modeling chemotaxis

2006 ◽  
Vol 2006 ◽  
pp. 1-23 ◽  
Author(s):  
Zhenbu Zhang
Keyword(s):  
Steady State ◽  
Global Existence ◽  
Global Solution ◽  
System Modeling ◽  
Reaction Diffusion ◽  
Steady States ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Steady State Solutions

We consider a reaction-diffusion system modeling chemotaxis, which describes the situation of two species of bacteria competing for the same nutrient. We use Moser-Alikakos iteration to prove the global existence of the solution. We also study the existence of nontrivial steady state solutions and their stability.

Global existence and large time behavior of solutions of a time fractional reaction diffusion system

2020 ◽  
Vol 23 (2) ◽  
pp. 390-407
Author(s):  
Ahmed Alsaedi ◽  
Bashir Ahmad ◽  
Mokhtar Kirane ◽  
Rafika Lassoued
Keyword(s):  
Global Existence ◽  
Large Time ◽  
Reaction Diffusion ◽  
Large Time Behavior ◽  
Positive Answer ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Time Behavior ◽  
Feedback Method ◽  
Fractional Reaction

AbstractIn this paper, it is proved that a time fractional reaction diffusion system with reaction terms of the Brusselator type admits a global solution by using the feedback method of F. Rothe [20]. Furthermore, some results on the large time behavior of the solutions are obtained. We give a positive answer to Problem 6 of the valuable paper of Gal and Warma [6].

ASYMPTOTIC EQUIVALENCE OF A REACTION-DIFFUSION SYSTEM TO THE CORRESPONDING SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

1995 ◽  
Vol 05 (06) ◽  
pp. 813-834 ◽  
Author(s):  
HIROKI HOSHINO ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Energy Method ◽  
Large Time ◽  
Reaction Diffusion ◽  
Large Time Behavior ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Time Behavior ◽  
Asymptotic Equivalence

Large time behavior of the solution to some simple reaction-diffusion system is studied. It is proved that the solution behaves like the solution to the corresponding system of ordinary differential equations as time goes to infinity. The proof is based on an energy method combined with the Lp−Lqestimate for the associated semigroup.

Stabilization of Dominant Structures in an Ionic Reaction-Diffusion System

1998 ◽  
Vol 63 (6) ◽  
pp. 761-769 ◽  
Author(s):  
Roland Krämer ◽  
Arno F. Münster
Keyword(s):  
Reaction Diffusion ◽  
Dominant Mode ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Local Perturbation ◽  
Dimensional System ◽  
One Dimensional ◽  
Brusselator Model ◽  
The One ◽  
Dominant Structure

We describe a method of stabilizing the dominant structure in a chaotic reaction-diffusion system, where the underlying nonlinear dynamics needs not to be known. The dominant mode is identified by the Karhunen-Loeve decomposition, also known as orthogonal decomposition. Using a ionic version of the Brusselator model in a spatially one-dimensional system, our control strategy is based on perturbations derived from the amplitude function of the dominant spatial mode. The perturbation is used in two different ways: A global perturbation is realized by forcing an electric current through the one-dimensional system, whereas the local perturbation is performed by modulating concentrations of the autocatalyst at the boundaries. Only the global method enhances the contribution of the dominant mode to the total fluctuation energy. On the other hand, the local method leads to simple bulk oscillation of the entire system.

scholarly journals The human EDAR 370V/A polymorphism affects tooth root morphology potentially through the modification of a reaction–diffusion system

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Keiichi Kataoka ◽  
Hironori Fujita ◽  
Mutsumi Isa ◽  
Shimpei Gotoh ◽  
Akira Arasaki ◽  
...  
Keyword(s):  
Molecular Mechanisms ◽  
Computational Study ◽  
Reaction Diffusion ◽  
Human Migration ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Tooth Root ◽  
Computed Tomography Images ◽  
Diffusion Dynamics ◽  
Root Morphogenesis

AbstractMorphological variations in human teeth have long been recognized and, in particular, the spatial and temporal distribution of two patterns of dental features in Asia, i.e., Sinodonty and Sundadonty, have contributed to our understanding of the human migration history. However, the molecular mechanisms underlying such dental variations have not yet been completely elucidated. Recent studies have clarified that a nonsynonymous variant in the ectodysplasin A receptor gene (EDAR370V/A; rs3827760) contributes to crown traits related to Sinodonty. In this study, we examined the association between theEDARpolymorphism and tooth root traits by using computed tomography images and identified that the effects of theEDARvariant on the number and shape of roots differed depending on the tooth type. In addition, to better understand tooth root morphogenesis, a computational analysis for patterns of tooth roots was performed, assuming a reaction–diffusion system. The computational study suggested that the complicated effects of theEDARpolymorphism could be explained when it is considered that EDAR modifies the syntheses of multiple related molecules working in the reaction–diffusion dynamics. In this study, we shed light on the molecular mechanisms of tooth root morphogenesis, which are less understood in comparison to those of tooth crown morphogenesis.

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