scholarly journals Reaction-diffusion systems with supercritical nonlinearities revisited

2021 ◽  
Author(s):  
Anna Kostianko ◽  
Chunyou Sun ◽  
Sergey Zelik
Keyword(s):  
Linear Part ◽  
Reaction Diffusion ◽  
Strong Solutions ◽  
Time Behavior ◽  
Exponential Attractors ◽  
Standard Case ◽  
Reaction Diffusion Systems ◽  
Supercritical Case ◽  
Diffusion Systems ◽  
Monotonicity Assumption

AbstractWe give a comprehensive study of the analytic properties and long-time behavior of solutions of a reaction-diffusion system in a bounded domain in the case where the nonlinearity satisfies the standard monotonicity assumption. We pay the main attention to the supercritical case, where the nonlinearity is not subordinated to the linear part of the equation trying to put as small as possible amount of extra restrictions on this nonlinearity. The properties of such systems in the supercritical case may be very different in comparison with the standard case of subordinated nonlinearities. We examine the global existence and uniqueness of weak and strong solutions, various types of smoothing properties, asymptotic compactness and the existence of global and exponential attractors.

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.

Initially Separated Reaction-Diffusion Systems

1994 ◽  
Vol 366 ◽  
Author(s):  
Haim Taitelbaum ◽  
Baruch Vilensky ◽  
Yong-Eun Lee Koo ◽  
Andrew Yen ◽  
Anna Lin ◽  
...  
Keyword(s):  
Reaction Front ◽  
Theoretical Study ◽  
Initial Density ◽  
Reaction Diffusion ◽  
Time Behavior ◽  
Reaction Diffusion Systems ◽  
System Parameters ◽  
Diffusion Systems ◽  
Reaction Constant ◽  
Asymptotic Time

ABSTRACTCharacteristics of the A + B → C reaction-diffusion system with initially separated components are studied theoretically and experimentally. When the reaction is slow, the two species will mix before reacting. This leads to a series of crossovers from a rich initial behavior to an asymptotic time behavior. The crossovers depend on the system parameters, such as the diffusion coefficients and initial densities of the two species. In this paper we review our recent studies on this system. We elaborate on a theoretical study of momentum effects, and then focus on theoretical explanations of two experimental phenomena: 1) Non-universal and non-monotonic motion of the reaction front center. The latter occurs when one of the reactants has larger diffusion coefficient but smaller initial density. 2) Existence of more than one front. This occurs when two different transformations of the same reactant (on one of the sides of the system), react with the reactant on the other side with a different reaction constant - the majority slowly, but the minority much faster.

Multiparameter bifurcation for some particular reaction–diffusion systems

1989 ◽  
Vol 112 (1-2) ◽  
pp. 135-143 ◽  
Author(s):  
J. Esquinas ◽  
J. López-Gómez
Keyword(s):  
Linear Operator ◽  
Linear Part ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Operator Polynomial ◽  
Reaction Diffusion Systems ◽  
Abstract Equation ◽  
Diffusion Systems ◽  
Multiparameter Bifurcation

SynopsisIn some cases, a reaction–diffusion system can be transformed into an abstract equation where the linear part is given by a polynomial of a linear operator, say Multiparameter bifurcation for this equation is considered as the coefficients of the operator polynomial in are varied.

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