Global dynamics below the ground state for the focusing semilinear Schrödinger equation with a linear potential
نویسندگان
چکیده
We study global dynamics of the solution to Cauchy problem for focusing semilinear Schrödinger equation with a linear potential on 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 complex-valued unknown function (t,x)∈I×R, I denotes maximal existence time interval u, V=V(x) non-negative and in L1(R)+L∞(R), p belongs so-called mass-supercritical case, i.e. p>5, H Hilbert space connected operator −∂x2+V called energy space. It well known that (NLSV) locally well-posed H. Our aim present paper behavior prove scattering result blow-up data u0 whose mass-energy less than ground state Q, Q=Q(x) unique radial positive stationary without potential:−Q″+Q=|Q|p−1Q,inH1(R). The similar NLS (V≡0), which invariant translation scaling transformation, one dimension was obtained by Akahori–Nawa. Lafontaine treated defocusing version (NLSV), is, replacement +|u|p−1u into −|u|p−1u, scatters as t→±∞ H1(R) an arbitrary Kenig-Merle's argument profile decomposition. However, method case cannot be applicable our because other hand, may negative case. To overcome this difficulty, we use variational argument. proof based Du–Wu–Zhang. difficulty lies deriving uniform bound functional related Virial Identity potential.
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 2021
ISSN: ['0022-247X', '1096-0813']
DOI: https://doi.org/10.1016/j.jmaa.2021.125291