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

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... ker 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.