scholarly journals Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems

2020 ◽  
Vol 18 (1) ◽  
pp. 1135-1149
Author(s):  
Yongqiang Fu ◽  
Lixu Yan
Keyword(s):  
Weak Solutions ◽  
Fractional Laplacian ◽  
Reaction Diffusion ◽  
Optimal Controls ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Stochastic Optimal Control Problem ◽  
Stochastic Reaction ◽  
Theoretical Results ◽  
Fractional Reaction

Abstract The aim of this paper is to investigate a class of nonlinear stochastic reaction-diffusion systems involving fractional Laplacian in a bounded domain. First, the existence and uniqueness of weak solutions are proved by using Galërkin’s method. Second, the existence of optimal controls for the corresponding stochastic optimal control problem is obtained. Finally, several examples are provided to demonstrate the theoretical results.

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.

Global Existence for Quadratic Systems of Reaction-Diffusion

2007 ◽  
Vol 7 (3) ◽  
Author(s):  
Laurent Desvillettes ◽  
Klemens Fellner ◽  
Michel Pierre ◽  
Julien Vovelle
Keyword(s):  
Global Existence ◽  
Weak Solutions ◽  
A Priori ◽  
Reaction Diffusion ◽  
Quadratic Systems ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
L Log L

AbstractWe prove global existence in time of weak solutions to a class of quadratic reaction-diffusion systems for which a Lyapounov structure of L log L-entropy type holds. The approach relies on an a priori dimension-independent L

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