Global well-posedness and nonlinear stability of a chemotaxis system modelling multiple sclerosis

2021 ◽  
pp. 1-31
Author(s):  
Laurent Desvillettes ◽  
Valeria Giunta ◽  
Jeff Morgan ◽  
Bao Quoc Tang
Keyword(s):  
Multiple Sclerosis ◽  
Nonlinear Stability ◽  
Coming Out ◽  
Stability Result ◽  
Reaction Diffusion ◽  
Strong Solutions ◽  
Reaction Diffusion Equations ◽  
System Modelling ◽  
Well Posedness ◽  
Chemotaxis System

We consider a system of reaction–diffusion equations including chemotaxis terms and coming out of the modelling of multiple sclerosis. The global existence of strong solutions to this system in any dimension is proved, and it is also shown that the solution is bounded uniformly in time. Finally, a nonlinear stability result is obtained when the chemotaxis term is not too big. We also perform numerical simulations to show the appearance of Turing patterns when the chemotaxis term is large.

Nonlinear stability and numerical simulations for a reaction–diffusion system modelling Allee effect on predators

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Florinda Capone ◽  
Maria Francesca Carfora ◽  
Roberta De Luca ◽  
Isabella Torcicollo
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Stability ◽  
Allee Effect ◽  
Complex Dynamics ◽  
Stability Result ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
System Modelling ◽  
Oscillatory Pattern

Abstract A reaction–diffusion system governing the prey–predator interaction with Allee effect on the predators, already introduced by the authors in a previous work is reconsidered with the aim of showing destabilization mechanisms of the biologically meaning equilibrium and detecting some aspects for the eventual oscillatory pattern formation. Extensive numerical simulations, depicting such complex dynamics, are shown. In order to complete the stability analysis of the coexistence equilibrium, a nonlinear stability result is shown.

scholarly journals Correction to: A local input-to-state stability result w.r.t. attractors of nonlinear reaction–diffusion equations

2021 ◽  
Author(s):  
Sergey Dashkovskiy ◽  
Oleksiy Kapustyan ◽  
Jochen Schmid
Keyword(s):  
Open Access ◽  
Stability Result ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlinear Reaction ◽  
Local Input

The article “A local input-to-state stability result w.r.t. attractors of nonlinear reaction–diffusion equations” written by Sergey Dashkovskiy, Oleksiy Kapustyan, and Jochen Schmid, was originally published online on 6 May 2020, without Open Access.

scholarly journals Time-dependent attractors for non-autonomous non-local reaction–diffusion equations

2018 ◽  
Vol 148 (5) ◽  
pp. 957-981
Author(s):  
Tomás Caraballo ◽  
Marta Herrera-Cobos ◽  
Pedro Marín-Rubio
Keyword(s):  
Existence And Uniqueness ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Strong Solutions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Pullback Attractors ◽  
Non Local ◽  
Bounded Sets

In this paper the existence and uniqueness of weak and strong solutions for a non-autonomous non-local reaction–diffusion equation is proved. Furthermore, the existence of minimal pullback attractors in the L2-norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition is established, along with some relationships between them. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 context. We also present new results in the autonomous framework that ensure the existence of global compact attractors as a particular case.

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