In 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 Q n C R", 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 Q n with appropriate boundary conditions for the Laplace operator, A, 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 €!" under suitable conditions. 1991 Mathematics subject classification (Amer. Math. Soc): primary 35P20, 34C29, 34C30, 35K57.

The g-Navier-Stokes equations in spatial dimension 2 were introduced by Roh aswith the continuity equationwhere g is a suitable smooth real valued function. Roh proved the existence of global solutions and the global attractor, for the spatial periodic and Dirichlet boundary conditions. Roh also proved that the global attractor A g of the g-Navier-Stokes equations converges (in the sense of upper continuity) to the global attractor A 1 of the Navier-Stokes equations as g → 1 in the proper sense.In this paper, we will estimate the dimension of the global attractor A g , for the spatial periodic and Dirichlet boundary conditions. Then, we will see that the upper bounds for the dimension of 437 the global attractors A g converge to the corresponding upper bounds for the global attractor A 1 as g → 1 in the proper sense.  2005 Elsevier Inc. All rights reserved.

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Abstract. We examine the existence of inertial manifold of a class of differential equations with particular boundary conditions.

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