On the Decay of Solutions to a Class of Defocusing Nls

We consider the following family of Cauchy problems: i∂tu = ∆u− u|u| , (t, x) ∈ R × R u(0) = φ ∈ H(R) where 0 < α < 4 d−2 for d ≥ 3 and 0 < α < ∞ for d = 1, 2. We prove that the L-norms of the solutions decay as t → ±∞, provided that 2 < r < 2d d−2 when d ≥ 3 and 2 < r < ∞ when d = 1, 2. In particular we extend previous results obtained in [5] for d ≥ 3 and in [8] for d = 1, 2, where the same decay results are proved under the extra assumption α > 4 d . This paper is devoted to the analysis of some asymptotic properties of solutions to the following family of defocusing NLS: (0.1) i∂tu = ∆u− u|u| , (t, x) ∈ R × R u(0) = φ ∈ H(R) where (0.2) 0 < α < 4 d− 2 for d ≥ 3 and 0 < α < ∞ for d = 1, 2. A lot of attention has been devoted in the literature to the Cauchy problem (0.1). In particular the questions of local and global well-posedness and scattering theory have been extensively studied. There exists an huge literature on the field and for this reason we cannot be exhaustive in the bibliography, however for the moment we would like to quote the book [1] for an extended description of the topics mentioned above and also for an extended bibliography. It is well–known from [4] (see also [6] for the more general question of unconditional uniqueness) that, under the assumptions (0.2) on α, for every initial data φ ∈ H(R) there exists a unique global solution u(t, x) ∈ C(R;H(R)) to (0.1). Moreover the global solutions of (0.1) satisfy the following conservation laws: (0.3) ‖u(t, x)‖L2(Rd) ≡ const ∀t ∈ R and (0.4) 1 2 ‖∇xu(t, x)‖ 2 L2(Rd) + 1 α+ 2 ‖u(t, x)‖ Lα+2(Rd) ≡ const ∀t ∈ R. The main contribution of this paper concerns the decay, in suitable Lebesgue spaces, of the global solutions to (0.1) as t → ±∞. 1