scholarly journals Attractors for Multivalued Impulsive Systems: Existence and Applications to Reaction-Diffusion System

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
S. Dashkovskiy ◽  
O. A. Kapustian ◽  
O. V. Kapustyan ◽  
N. V. Gorban
Keyword(s):  
Initial Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Impulsive Systems ◽  
Uniform Attractor ◽  
Infinite Dimensional ◽  
Long Time

In this paper, we develop a general approach to investigate limit dynamics of infinite-dimensional dissipative impulsive systems whose initial conditions do not uniquely determine their long time behavior. Based on the notion of an uniform attractor, we show how to describe limit behavior of such complex systems with the help of properties of their components. More precisely, we prove the existence of the uniform attractor for an impulsive multivalued system in terms of properties of nonimpulsive semiflow and impulsive parameters. We also give an application of these abstract results to the impulsive reaction-diffusion system without uniqueness.

scholarly journals Asymptotic Analysis for a nonlinear Reaction-Diffusion System Modeling an Infectious Disease

2021 ◽  
Author(s):  
Hong-Ming Yin ◽  
Jun Zou
Keyword(s):  
Infectious Disease ◽  
System Modeling ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Time Behavior ◽  
Strongly Coupled ◽  
Nonlinear Reaction ◽  
Open Questions ◽  
Long Time

In this paper we study a nonlinear reaction-diffusion system which models an infectious disease caused by bacteria such as cholera. One of the features in this model is that a certain portion of the recovered human hosts lost a lifetime immunity and could be infected again. Another feature in the model is that the mobility for each species is assumed to be dependent upon location and time. We also assume that the whole group is susceptible with the bacteria. This leads to a strongly coupled nonlinear reaction-diffusion system. We prove that the nonlinear system has a unique solution globally in any space dimension under some natural conditions on known parameters and functions. Moreover, the long-time behavior and stability analysis for the solution are carried out rigorously. In particular, we characterize the precise conditions on variable parameters about the stability or instability for all steady-state solutions. These results obtained in this paper answered several open questions raised in the previous literature

scholarly journals Necessary conditions for local and global existence to a reaction-diffusion system with fractional derivatives

2006 ◽  
Vol 2006 ◽  
pp. 1-8 ◽  
Author(s):  
Kamel Haouam ◽  
Mourad Sfaxi
Keyword(s):  
Global Existence ◽  
Fractional Derivatives ◽  
Necessary Conditions ◽  
Initial Conditions ◽  
Reaction Diffusion ◽  
Single Equation ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Existence Of A Solution

We give some necessary conditions for local and global existence of a solution to reaction-diffusion system of type (FDS) with temporal and spacial fractional derivatives. As in the case of single equation of type (STFE) studied by M. Kirane et al. (2005), we prove that these conditions depend on the behavior of initial conditions for large|x|.

Long-time behavior of solutions and chaos in reaction-diffusion equations

2017 ◽  
Vol 99 ◽  
pp. 91-100 ◽  
Author(s):  
Kamal N. Soltanov ◽  
Anatolij K. Prykarpatski ◽  
Denis Blackmore
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time

LONG-TIME BEHAVIOR OF SOLUTIONS TO NONLINEAR REACTION DIFFUSION EQUATIONS INVOLVING L1 DATA

2012 ◽  
Vol 14 (01) ◽  
pp. 1250007 ◽  
Author(s):  
CHENGKUI ZHONG ◽  
WEISHENG NIU
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Smooth Bounded Domain ◽  
Bootstrap Argument ◽  
Nonlinear Reaction ◽  
L1 Data ◽  
Long Time

In this paper we consider the long-time behavior of solutions to nonlinear reaction diffusion equations involving L1 data, [Formula: see text] where Ω is a smooth bounded domain and u0, g ∈ L1(Ω). Using a decomposition technique combined with a bootstrap argument we establish some uniform regularity results on the solutions, by which we prove that the solution semigroup generated by the problem above possesses a global attractor [Formula: see text] in L1(Ω). Moreover, we obtain that the attractor is actually invariant, compact in [Formula: see text], q < max {N/(N-1), (2p-2)/p}, and attracts every bounded subset of L1(Ω) in the norm of [Formula: see text], 1 ≤ r < ∞.

Sign in / Sign up

Export Citation Format

Share Document