A simple proof of scattering for the intercritical inhomogeneous NLS. (English) Zbl 1483.35219
Summary: We adapt the argument of B. Dodson and the author [ibid. 145, No. 11, 4859–4867 (2017; Zbl 1373.35287)] to give a simple proof of scattering below the ground state for the intercritical inhomogeneous nonlinear Schrödinger equation. The decaying factor in the nonlinearity obviates the need for a radial assumption.
Citations:
Zbl 1373.35287References:
| [1] | Arora, Anudeep Kumar; Dodson, Benjamin; Murphy, Jason, Scattering below the ground state for the \(2d\) radial nonlinear Schr\"{o}dinger equation, Proc. Amer. Math. Soc., 148, 4, 1653-1663 (2020) · Zbl 1435.35341 · doi:10.1090/proc/14824 |
| [2] | Campos, Luccas, Scattering of radial solutions to the inhomogeneous nonlinear Schr\"{o}dinger equation, Nonlinear Anal., 202, 112118, 17 pp. (2021) · Zbl 1452.35179 · doi:10.1016/j.na.2020.112118 |
| [3] | CFGM M. Cardoso, L. G. Farah, C. M. Guzm\'an, and J. Murphy, Scattering below the ground state for the intercritical non-radial inhomogeneous NLS, Preprint, 2007.06165, 2020. |
| [4] | Cazenave, Thierry, Semilinear Schr\"{o}dinger equations, Courant Lecture Notes in Mathematics 10, xiv+323 pp. (2003), New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI · Zbl 1055.35003 · doi:10.1090/cln/010 |
| [5] | Dodson, Benjamin; Murphy, Jason, A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, Proc. Amer. Math. Soc., 145, 11, 4859-4867 (2017) · Zbl 1373.35287 · doi:10.1090/proc/13678 |
| [6] | Farah, Luiz G., Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schr\"{o}dinger equation, J. Evol. Equ., 16, 1, 193-208 (2016) · Zbl 1339.35287 · doi:10.1007/s00028-015-0298-y |
| [7] | Farah, Luiz Gustavo; Guzm\'{a}n, Carlos M., Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schr\"{o}dinger equation, J. Differential Equations, 262, 8, 4175-4231 (2017) · Zbl 1362.35284 · doi:10.1016/j.jde.2017.01.013 |
| [8] | Guzm\'{a}n, Carlos M., On well posedness for the inhomogeneous nonlinear Schr\"{o}dinger equation, Nonlinear Anal. Real World Appl., 37, 249-286 (2017) · Zbl 1375.35486 · doi:10.1016/j.nonrwa.2017.02.018 |
| [9] | Hunt, Richard A., On \(L(p,\,q)\) spaces, Enseign. Math. (2), 12, 249-276 (1966) · Zbl 0181.40301 |
| [10] | Kenig, Carlos E.; Merle, Frank, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schr\"{o}dinger equation in the radial case, Invent. Math., 166, 3, 645-675 (2006) · Zbl 1115.35125 · doi:10.1007/s00222-006-0011-4 |
| [11] | MMZ C. Miao, J. Murphy, and J. Zheng, Scattering for the non-radial inhomogeneous NLS, Preprint, 1912.01318, 2019, to appear in Math. Res. Lett. |
| [12] | Ogawa, Takayoshi; Tsutsumi, Yoshio, Blow-up of \(H^1\) solution for the nonlinear Schr\"{o}dinger equation, J. Differential Equations, 92, 2, 317-330 (1991) · Zbl 0739.35093 · doi:10.1016/0022-0396(91)90052-B |
| [13] | O’Neil, Richard, Convolution operators and \(L(p,\,q)\) spaces, Duke Math. J., 30, 129-142 (1963) · Zbl 0178.47701 |
| [14] | Stein, Elias M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, xiv+290 pp. (1970), Princeton University Press, Princeton, N.J. · Zbl 0207.13501 |
| [15] | Strauss, Walter A., Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55, 2, 149-162 (1977) · Zbl 0356.35028 |
| [16] | Tao, Terence, On the asymptotic behavior of large radial data for a focusing non-linear Schr\"{o}dinger equation, Dyn. Partial Differ. Equ., 1, 1, 1-48 (2004) · Zbl 1082.35144 · doi:10.4310/DPDE.2004.v1.n1.a1 |
| [17] | XZ C. Xu and T. Zhao, A remark on the scattering theory for the 2d radial focusing INLS, Preprint, 1908.00743, 2019. |
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