TY - JOUR
T1 - On Hartree equations with derivatives
AU - Cho, Yonggeun
AU - Lee, Sanghyuk
AU - Ozawa, Tohru
N1 - Funding Information:
The authors would like to thank the referees for their careful reading and helpful comments. The first author was supported by the Korea Research Foundation Grant funded by the Korean Government ( KRF-2008-313-C00065 ). The second author was supported by NRF grant 2009-0093131 (Korea).
PY - 2011/3/15
Y1 - 2011/3/15
N2 - We consider the Cauchy problem of two types of Hartree equations with exchangecorrelation correction terms: {iut - Δu = V k(u)u in ℝ1+n, k = 1,2,u(0) = φ in ℝn, n ≥ 1, where V1(u) = |x|-γ*1|u|2 + λ2|∇u| 2),V2(u) = |x|-γ *∥∇|δu|2). We establish the well-posedness of Cauchy problems and show the smoothing effect of solutions for each 0 < γ < n and 0 ≤ δ ≤ 1.
AB - We consider the Cauchy problem of two types of Hartree equations with exchangecorrelation correction terms: {iut - Δu = V k(u)u in ℝ1+n, k = 1,2,u(0) = φ in ℝn, n ≥ 1, where V1(u) = |x|-γ*1|u|2 + λ2|∇u| 2),V2(u) = |x|-γ *∥∇|δu|2). We establish the well-posedness of Cauchy problems and show the smoothing effect of solutions for each 0 < γ < n and 0 ≤ δ ≤ 1.
KW - Angular regularity
KW - Hartree equations with derivatives
KW - Smoothing effect
KW - Well-posedness
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U2 - 10.1016/j.na.2010.11.015
DO - 10.1016/j.na.2010.11.015
M3 - Article
AN - SCOPUS:79951813303
SN - 0362-546X
VL - 74
SP - 2094
EP - 2108
JO - Nonlinear Analysis, Theory, Methods and Applications
JF - Nonlinear Analysis, Theory, Methods and Applications
IS - 6
ER -