ON MINIMAL SETS OF SCALAR PARABOLIC EQUATIONS WITH SKEW-PRODUCT STRUCTURES

Skew-product semi-flow Pi(t):X x Y --> X x Y which is generated by GR APHICS is considered, where X is an appropriate subspace of H-2(0, 1) , (Y, R) is a minimal flow with compact phase space. It is shown that a minimal set E subset of X x Y of Pi(t) is an almost 1-1 extension of Y, that is, set Y-0 = {y is an element of Y\card(E subset of P-1(y)) = 1} is a residual subset of Y, where P:X x Y --> Y is the natural pro jection. Consequently, if (Y, R) is almost periodic minimal, then any minimal set E subset of X x Y of Pi(t) is an almost automorphic minima l set. It is also proved that dynamics of Pi(t) is closed in the categ ory of almost automorphy, that is, a minimal set E subset of X x Y of Pi(t) is almost automorphic minimal if and only if (Y, R) is almost au tomorphic minimal. Asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed. Examples of nonal most periodic almost automorphic minimal sets are provided.