Approximate inertial manifolds for reaction-diffusion equations in high space dimension

1989 ◽  
Vol 1 (3) ◽  
pp. 245-267 ◽  
Author(s):  
Martine Marion
Keyword(s):  
Space Dimension ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Inertial Manifolds ◽  
Approximate Inertial Manifolds ◽  
High Space

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.

scholarly journals Approximate inertial manifolds for nonlinear parabolic equations via steady-state determining mapping

1995 ◽  
Vol 18 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Yuncheng You
Keyword(s):  
Steady State ◽  
Evolution Equations ◽  
Fractional Power ◽  
Reaction Diffusion ◽  
Nonlinear Parabolic Equations ◽  
Reaction Diffusion Equations ◽  
Inertial Manifolds ◽  
Axially Symmetric ◽  
Nonlinear Parabolic ◽  
Approximate Inertial Manifolds

For nonlinear parabolic evolution equations, it is proved that, under the assumptions of local Lipschitz continuity of nonlinearity and the dissipativity of semiflows, there exist approximate inertial manifolds (AIM) in the energy space and that the approximate inertial manifolds are constructed as the graph of the steady-state determining mapping based on the spectral decomposition. It is also shown that the thickness of the exponentially attracting neighborhood of the AIM converges to zero at a fractional power rate as the dimension of the AIM increases. Applications of the obtained results to Burgers' equation, higher dimensional reaction-diffusion equations, 2D Ginzburg-Landau equations, and axially symmetric Kuramoto-Sivashinsky equations in annular domains are included.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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