Global dynamics below the ground state for the focusing semilinear Schrödinger equation with a linear potential

Abstract

We study global dynamics of the solution to the Cauchy problem for the focusing semilinear Schrödinger equation with a linear potential on the real line R:(NLSV){i∂tu+∂x2u−Vu+|u|p−1u=0,(t,x)∈I×R,u(0)=u0∈H, where u=u(t,x) is a complex-valued unknown function of (t,x)∈I×R, I denotes the maximal existence time interval of u, V=V(x) is non-negative and in L1(R)+L∞(R), p belongs to the so-called mass-supercritical case, i.e. p>5, and H is a Hilbert space connected to the Schrödinger operator −∂x2+V and is called energy space. It is well known that (NLSV) is locally well-posed in H. Our aim in the present paper is to study global behavior of the solution and prove a scattering result and a blow-up result for (NLSV) with the data u0 whose mass-energy is less than that of the ground state Q, where the function Q=Q(x) is the unique radial positive solution to the stationary Schrödinger equation without the potential:−Q″+Q=|Q|p−1Q,inH1(R). The similar result for NLS without potential (V≡0), which is invariant of translation and scaling transformation, in one space dimension was obtained by Akahori–Nawa. Lafontaine treated the defocusing version of (NLSV), that is, (NLSV) with a replacement of +|u|p−1u into −|u|p−1u, and prove that the solution scatters as t→±∞ in H1(R) for an arbitrary data in H1(R) by Kenig-Merle's argument with a profile decomposition. However, the method to the defocusing case cannot be applicable to our focusing case because the energy is positive in the defocusing case, on the other hand, the energy may be negative in the focusing case. To overcome this difficulty, we use a variational argument. Our proof of the blow-up result is based on the argument of Du–Wu–Zhang. The difficulty of our case lies in deriving a uniform bound of a functional related to Virial Identity because of existence of the potential.