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

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 (...

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 p...

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In this paper we establish asymptotic (biasymptotic) equivalence between spaces of solutions of a given linear homogeneous system and a perturbed system. The perturbations are of either linear or weakly linear characters. Existence of a homeomorphism between subspaces of almost periodic and asymptotically (biasymptotically) almost periodic solutions is also obtained.