Asymptotic Almost Periodicity of Scalar Parabolic Equations with Almost Periodic Time Dependence

1 1. Introduction This paper is devoted to the study of asymptotic almost periodicity of bounded solutions for the following time almost periodic one dimensional scalar parabolic equation:    u t = u xx + f (t, x, u, u x), t > 0, 0 < x < 1, u(t, 0) = u(t, 1) = 0, t > 0, (1.1) where f : IR 1 × [0, 1] × IR 1 × IR 1 → IR 1 is a C 2 function, and f (t, x, u, p) with all its partial derivatives (up to order 2) are (Bohr) almost periodic in t uniformly for other variables in compact subsets. Denote by H(f) the hull of f in compact open topology and let X α be a fractional power space associated with the operator u → −u xx : H 2 0 (0, 1) → L 2 (0, 1) that satisfies X α → C 1 [0, 1] (that is, X α is compact embedded in C 1 [0, 1]). Equation (1.1) generates a (local) skew product semiflow Π t on X α × H(f) (see section 2) as follows: Π t (U, g) = (u(t, ·, U, g), g · t), (1.2) where g · t is the flow on H(f) defined by time translations (H(f) is therefore almost periodic minimal under this flow, see [9], [20]), u(t, ·, U, g) is the solution of    u t = u xx + g(t, x, u, u x), t > 0, 0 < x < 1, u(t, 0) = u(t, 1) = 0, t > 0 (1.3) g with u(0, ·, U, g) = U (·). We shall study the asymptotic almost periodicity for a positively bounded motion Π t (U, g) of (1.2) by investigating its ω-limit set ω(U, g) (the set of all accumulation points of Π t (U, g) as t goes to infinite) since it has been shown in [22] that Π t (U, g) is asymptotically almost periodic if and only if ω(U, g) is an almost periodic extension of H(f) (namely, a 1-cover of H(f)). We note that for (1.2), an ω-limit set is an almost periodic extension of H(f) if and only if it is almost periodic minimal ([22]). For a time periodic parabolic equation of form (1.1), it is known that any bounded solution is asymptotically periodic, that …