scholarly journals On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains

2003 ◽  
Vol 154 (3) ◽  
pp. 253-275 ◽  
Author(s):  
M. Prizzi ◽  
K. P. Rybakowski
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Inertial Manifolds ◽  
High Dimensional ◽  
Thin Domains

scholarly journals An extension of the principle of spatial averaging for inertial manifolds

1999 ◽  
Vol 66 (1) ◽  
pp. 125-142 ◽  
Author(s):  
Hyukjin Kwean
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Inertial Manifolds ◽  
Inertial Manifold ◽  
Specific Class ◽  
Spatial Averaging ◽  
The Laplace Operator

AbstractIn this paper we extend a theorem of Mallet-Paret and Sell for the existence of an inertial manifold for a scalar-valued reaction diffusion equation to new physical domains ωn ⊂ Rn, n = 2,3. For their result the Principle of Spatial Averaging (PSA), which certain domains may possess, plays a key role for the existence of an inertial manifold. Instead of the PSA, we define a weaker PSA and prove that the domains φn with appropriate boundary conditions for the Laplace operator, δ, satisfy a weaker PSA. This weaker PSA is enough to ensure the existence of an inertial manifold for a specific class of scalar-valued reaction diffusion equations on each domain ωn under suitable conditions.

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