Backstepping-based adaptive error feedback regulator design for one-dimensional reaction-diffusion equation

2020 ◽  
Vol 484 (1) ◽  
pp. 123666
Author(s):  
Wei-Wei Liu ◽  
Wei Guo ◽  
Jun-Min Wang
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Error Feedback ◽  
One Dimensional ◽  
Feedback Regulator

scholarly journals A reaction-diffusion equation on a thin L-shaped domain

1995 ◽  
Vol 125 (2) ◽  
pp. 283-327 ◽  
Author(s):  
Jack K. Hale ◽  
Geneviève Raugel
Keyword(s):  
Diffusion Equation ◽  
Lower Semicontinuity ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Global Attractors ◽  
Reaction Diffusion Equation ◽  
Limit Equation ◽  
One Dimensional ◽  
Shaped Domain

We consider a dissipative reaction–diffusion equation on a thin L-shaped domain (with the thinness measured by a parameter ε); we determine the limit equation for ε = 0 and prove the upper semicontinuity of the global attractors at ε = 0. We also state a lower semicontinuity result. When the limit equation is one-dimensional, we prove convergence of any orbit to a singleton.

Nucleation Rate of a Localized Structure in a Reaction-Diffusion System

1993 ◽  
Vol 48 (5-6) ◽  
pp. 636-638 ◽  
Author(s):  
T. Christen
Keyword(s):  
Diffusion Equation ◽  
Nucleation Rate ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Reaction Diffusion Equation ◽  
One Dimensional ◽  
Uniform State

Abstract We derive the nucleation rate of a localized structure of a one-dimensional, nonlocal, bistable reaction diffusion equation near instability of the uniform state.

THE QUENCHING OF SOLUTIONS OF A REACTION–DIFFUSION EQUATION WITH FREE BOUNDARIES

2016 ◽  
Vol 94 (1) ◽  
pp. 110-120 ◽  
Author(s):  
NINGKUI SUN
Keyword(s):  
Initial Data ◽  
Diffusion Equation ◽  
Compact Subset ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Free Boundaries ◽  
One Dimensional ◽  
All Solutions ◽  
Quenching Set ◽  
Stefan Condition

This paper concerns the quenching phenomena of a reaction–diffusion equation $u_{t}=u_{xx}+1/(1-u)$ in a one dimensional varying domain $[g(t),h(t)]$, where $g(t)$ and $h(t)$ are two free boundaries evolving by a Stefan condition. We prove that all solutions will quench regardless of the choice of initial data, and we also show that the quenching set is a compact subset of the initial occupying domain and that the two free boundaries remain bounded.

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