首页|Small Data Global Well-Posedness and Scattering for the Inhomogeneous Nonlinear Schrodinger Equation in H-s(R-n)
Small Data Global Well-Posedness and Scattering for the Inhomogeneous Nonlinear Schrodinger Equation in H-s(R-n)
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NSTL
We consider the Cauchy problem for the inhomogeneous nonlinear Schrodinger (INLS) equation iu(t) + Delta u = vertical bar x vertical bar(-b)f (u), u(0) = u(0) is an element of H-s(R-n), where 0 < s < min{n, n/2 + 1}, 0 < b min{2, n - s, 1 + n-2s/2} and f (u) is a nonlinear function 2 that behaves like lambda vertical bar u vertical bar(sigma) u with lambda is an element of C and sigma > 0, We prove that the Cauchy problem of the INLS equation is globally well-posed in H-s(R-n) if the initial data is sufficiently small and sigma(0) < sigma < sigma(s), where sigma(0) = 4-2b/n and sigma(s) = 4-2b/n-2s if s < n/2, sigma(s) = infinity if s >= n/2. Our global well-posedness result improves the one of Guzman [Nonlinear Anal. Real World Appl. 37 (2017), 249-286] by extending the validity of s and b. In addition, we also have the small data scattering result.
NSTL
