scholarly journals A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions

2015 ◽  
Vol 61 (1) ◽  
pp. 59-78 ◽  
Author(s):  
Monica-Dana Burlică ◽  
Daniela Roşu
Keyword(s):  
Fixed Point ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Diffusion System ◽  
Reaction Diffusion Systems ◽  
Nonlocal Initial Conditions ◽  
Diffusion Systems ◽  
Compactness Arguments ◽  
Fixed Point Techniques

Abstract We consider an abstract nonlinear multi-valued reaction-diffusion system with delay and, using some compactness arguments coupled with metric fixed point techniques, we prove some sufficient conditions for the existence of at least one C0-solution.

Large-time dynamics in complex networks of reaction–diffusion systems applied to a panic model

2019 ◽  
Vol 84 (5) ◽  
pp. 974-1000
Author(s):  
Guillaume Cantin ◽  
M A Aziz-Alaoui ◽  
Nathalie Verdière
Keyword(s):  
Large Time ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Energy Estimates ◽  
Invariant Region ◽  
Exponential Attractors ◽  
Reaction Diffusion Systems ◽  
Time Dynamics ◽  
Diffusion Systems ◽  
Theoretical Results

Abstract This paper is devoted to the analysis of the asymptotic behaviour of a complex network of reaction–diffusion systems for a geographical model, which was proposed recently, in order to better understand behavioural reactions of individuals facing a catastrophic event. After stating sufficient conditions for the problem to admit a positively invariant region, we establish energy estimates and prove the existence of a family of exponential attractors. We explore the influence of the size of the network on the nature of those attractors, in correspondence with the geographical background. Numerical simulations illustrate our theoretical results and show the various possible dynamics of the problem.

scholarly journals Stability of reaction–diffusion systems with stochastic switching

2019 ◽  
Vol 24 (3) ◽  
pp. 315-331 ◽  
Author(s):  
Lijun Pan ◽  
Jinde Cao ◽  
Ahmed Alsaedi
Keyword(s):  
Direct Method ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Fubini Theorem ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Stochastic Switching ◽  
The Stability ◽  
Independent And Identically Distributed

In this paper, we investigate the stability for reaction systems with stochastic switching. Two types of switched models are considered: (i) Markov switching and (ii) independent and identically distributed switching. By means of the ergodic property of Markov chain, Dynkin formula and Fubini theorem, together with the Lyapunov direct method, some sufficient conditions are obtained to ensure that the zero solution of reaction–diffusion systems with Markov switching is almost surely exponential stable or exponentially stable in the mean square. By using Theorem 7.3 in [R. Durrett, Probability: Theory and Examples, Duxbury Press, Belmont, CA, 2005], we also investigate the stability of reaction–diffusion systems with independent and identically distributed switching. Meanwhile, an example with simulations is provided to certify that the stochastic switching plays an essential role in the stability of systems.

ABRAHAMS–TSUNETO REACTION–DIFFUSION SYSTEM IN SUPERCONDUCTIVITY AND SYMBOLIC COMPUTATION

2001 ◽  
Vol 12 (10) ◽  
pp. 1417-1423 ◽  
Author(s):  
YI-TIAN GAO ◽  
BO TIAN ◽  
GUANG-MEI WEI
Keyword(s):  
Symbolic Computation ◽  
Reaction Diffusion ◽  
Analytic Solutions ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Ginzburg Landau ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Exact Analytic Solutions ◽  
Self Consistent

The reaction–diffusion systems represent various problems in the real world. For the Abrahams–Tsuneto reaction–diffusion system arising in superconductivity, we perform computerized symbolic computation and find its new exact analytic solutions, which are solitonic. We see the possibility that by way of the shock waves, the self–consistent superconducting interaction drives the Ginzburg–Landau order parameter, which might be observable.

scholarly journals Controllability and Observability of Nonautonomous Riesz-Spectral Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Sutrima Sutrima ◽  
Christiana Rini Indrati ◽  
Lina Aryati
Keyword(s):  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Dynamical Behaviour ◽  
Transport Reaction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Semigroup Approach ◽  
Sturm Liouville ◽  
Controllability And Observability

There are many industrial and biological reaction diffusion systems which involve the time-varying features where certain parameters of the system change during the process. A part of the transport-reaction phenomena is often modelled as an abstract nonautonomous equation generated by a (generalized) Riesz-spectral operator on a Hilbert space. The basic problems related to the equations are existence of solutions of the equations and how to control dynamical behaviour of the equations. In contrast to the autonomous control problems, theory of controllability and observability for the nonautonomous systems is less well established. In this paper, we consider some relevant aspects regarding the controllability and observability for the nonautonomous Riesz-spectral systems including the Sturm-Liouville systems using a C0-quasi-semigroup approach. Three examples are provided. The first is related to sufficient conditions for the existence of solutions and the others are to confirm the approximate controllability and observability of the nonautonomous Riesz-spectral systems and Sturm-Liouville systems, respectively.

scholarly journals Exact Controllability for Hilfer Fractional Differential Inclusions Involving Nonlocal Initial Conditions

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Jun Du ◽  
Wei Jiang ◽  
Denghao Pang ◽  
Azmat Ullah Khan Niazi
Keyword(s):  
Fixed Point ◽  
Differential Inclusions ◽  
Measure Of Noncompactness ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Exact Controllability ◽  
Fractional Differential Inclusions ◽  
Nonlocal Initial Conditions ◽  
Fractional Differential System ◽  
Fractional Differential

The exact controllability results for Hilfer fractional differential inclusions involving nonlocal initial conditions are presented and proved. By means of the multivalued analysis, measure of noncompactness method, fractional calculus combined with the generalized Mo¨nch fixed point theorem, we derive some sufficient conditions to ensure the controllability for the nonlocal Hilfer fractional differential system. The results are new and generalize the existing results. Finally, we talk about an example to interpret the applications of our abstract results.

Replicating Spots in Reaction-Diffusion Systems

1997 ◽  
Vol 07 (05) ◽  
pp. 1149-1158 ◽  
Author(s):  
Kyoung J. Lee ◽  
Harry L. Swinney
Keyword(s):  
Cell Division ◽  
Numerical Simulations ◽  
Continuous Process ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Diffusion System ◽  
One Dimension ◽  
Reaction Diffusion Systems ◽  
Gel Layer ◽  
Diffusion Systems

We review the phenomenon of replicating spots in reaction-diffusion systems and discuss the mechanism of replication. This phenomenon was discovered in recent experiments on a ferrocyanide-iodate-sulfite reaction-diffusion system. Patterns form in a thin gel layer that is in contact with a continuously fed stirred reservoir. Patterns of spots are observed to undergo a continuous process of growth and multiplication through cell division and death through overcrowding. A similar phenomenon is also found in numerical simulations in one dimension on a four-species model of the ferrocyanide-iodate-sulfite reaction and in simulations in two dimensions of simpler two-species reaction-diffusion models: Gray–Scott model by J. Pearson and FitzHugh–Nagumo model by A. Hagberg and E. Meron.

ON A NONLOCAL EIGENVALUE PROBLEM AND ITS APPLICATIONS TO POINT-CONDENSATIONS IN REACTION–DIFFUSION SYSTEMS

2000 ◽  
Vol 10 (06) ◽  
pp. 1485-1496 ◽  
Author(s):  
JUNCHENG WEI
Keyword(s):  
Eigenvalue Problem ◽  
Single Point ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Stability And Instability ◽  
Nonlocal Eigenvalue Problem

We consider a nonlocal eigenvalue problem which arises in the study of stability of point-condensation solutions in the Gierer–Meinhardt system and generalized Gray–Scott system. We give some sufficient conditions for stability and instability. The conditions are new and can be applied to the study of stability of single point-condensation solutions.

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