  In this paper we study a family of semilinear reaction-diffusion equations on
thin spatial domains, lying close to a lower dimensional submanifold $M$. As
the thickness tends to zero, the domains collapse onto (a subset of) $M$. As it
was proved in a previous paper (M. Prizzi, M. Rinaldi and K. P. Rybakowski,
Curved thin domains and parabolic equations, Stud. Math. 151), the above family
has a limit equation, which is an abstract semilinear parabolic equation
defined on a certain abstract limit phase space. One of the objectives of this
paper is to give more manageable characterizations of the limit phase space.
Under additional hypotheses, we also give a simple description of the limit
equation. If, in addition, $M$ is a sphere and the nonlinearity of the above
equations is dissipative, we prove that, if the thickness is small enough, the
corresponding equation possesses an inertial manifold, i.e. an invariant
manifold containing the attractor of the equation. We thus obtain the existence
of inertial manifolds for reaction-diffusion equations on certain classes of
thin domains of genuinely high dimension.
