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.

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 A Nonlocal Eigenvalue Problem and the Stability of Spikes for Reaction–Diffusion Systems with Fractional Reaction Rates

2003 ◽  
Vol 13 (06) ◽  
pp. 1529-1543 ◽  
Author(s):  
Juncheng Wei ◽  
Matthias Winter
Keyword(s):  
Eigenvalue Problem ◽  
Reaction Diffusion ◽  
Reaction Rates ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
The Stability ◽  
Nonlocal Eigenvalue Problem ◽  
Fractional Reaction

We consider a nonlocal eigenvalue problem which arises in the study of stability of spike solutions for reaction–diffusion systems with fractional reaction rates such as the Sel'kov model, the Gray–Scott system, the hypercycle of Eigen and Schuster, angiogenesis, and the generalized Gierer–Meinhardt system. We give some sufficient and explicit conditions for stability by studying the corresponding nonlocal eigenvalue problem in a new range of parameters.

scholarly journals On Classical Solutions for A Kuramoto–Sinelshchikov–Velarde-Type Equation

Algorithms ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 77 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Thermal Conduction ◽  
Type Equation ◽  
Reaction Diffusion ◽  
Classical Solutions ◽  
Well Posedness ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
The Cauchy Problem ◽  
The Stability ◽  
Plane Flame

The Kuramoto–Sinelshchikov–Velarde equation describes the evolution of a phase turbulence in reaction-diffusion systems or the evolution of the plane flame propagation, taking into account the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

Different Types of Instabilities and Complex Dynamics in Reaction-Diffusion Systems With Fractional Derivatives

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
Vasyl Gafiychuk ◽  
Bohdan Datsko
Keyword(s):  
Complex Dynamics ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Different Types ◽  
Linearized System ◽  
The Stability ◽  
Nonlinear Solutions ◽  
Activator Inhibitor ◽  
Fractional Reaction

In this article we analyze conditions for different types of instabilities and complex dynamics that occur in nonlinear two-component fractional reaction-diffusion systems. It is shown that the stability of steady state solutions and their evolution are mainly determined by the eigenvalue spectrum of a linearized system and the fractional derivative order. The results of the linear stability analysis are confirmed by computer simulations of the FitzHugh-Nahumo-like model. On the basis of this model, it is demonstrated that the conditions of instability and the pattern formation dynamics in fractional activator- inhibitor systems are different from the standard ones. As a result, a richer and a more complicated spatiotemporal dynamics takes place in fractional reaction-diffusion systems. A common picture of nonlinear solutions in time-fractional reaction-diffusion systems and illustrative examples are presented. The results obtained in the article for homogeneous perturbation have also been of interest for dynamical systems described by fractional ordinary differential equations.

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 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.

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.

Sign in / Sign up

Export Citation Format

Share Document