Dynamics of Almost Periodic Scalar Parabolic Equations

The current paper is devoted to study of the asymptotic behavior of bounded solutions for the following type of parabolic equation: u t = u xx + f (t, x, u, u x), t > 0, 0 < x < 1, (1.1) with the boundary conditions: βu(t, 0) + (1 − β)u x (t, 0) = 0, βu(t, 1) + (1 − β)u x (t, 1) = 0, t > 0, (1.2) where β = 0 or 1, f : IR 1 × [0, 1] × IR 1 × IR 1 → IR 1 is C 2 , and f (t, x, u, p) with all its partial derivatives (up to order 2) are almost periodic in t uniformly for (x, u, p) in compact subsets. To carry out our study for the nonautonomous equation (1.1)-(1.2), we define a dy-namical system associated to it in the following way. Let C = C(IR 1 ×[0, 1]×IR 1 ×IR 1 , IR 1) be the space of continuous functions F : IR 1 × [0, 1] × IR 1 × IR 1 → IR 1. Give C the compact open topology, that is, the topology of uniform convergence on compact subsets. It follows from classical topological dynamical system theory ([26]) that the time translation (F, t) → F t : F t (s, x, u, p) = F (t + s, x, u, p) defines a flow on C, and the hull of f , H(f) = cl{f t |t ∈ IR 1 } is an almost periodic minimal set (that is, H(f) is minimal and each motion in H(f) is almost periodic). Furthermore, each g ∈ H(f) is also a C 2 function (see [17]). By introducing the hull H(f), (1.1)-(1.2) gives rise to a family of equations associated to each g ∈ H(f),    u t = u xx + g(t, x, u, u x), t > 0, 0 < x < 1, βu(t, 0) + (1 − β)u x (t, 0) = 0, βu(t, 1) + (1 − β)u x (t, 1) = 0, t > 0. defines a (local) skew product semiflow Π t on X × H(f) : Π t (U, g) = (u(t, ·, U, g), g · t), t > 0, (1.4) 2 where g · t is the flow on H(f) defined by time translations. In the terminology of the (local) skew …