Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials

References

• Ginibre , J and Velo , G . 1979 . On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case . J. Funct. Anal. , 32 : 1 – 32 .
   Web of Science ®Google Scholar
• Kato , T . 1987 . On nonlinear Schrödinger equations . Ann. Inst. H. Poincaré, Phys. Théor. , 46 : 113 – 129 .
   Web of Science ®Google Scholar
• Tsutsumi , Y . 1987 . L 2-solutions for nonlinear Schrödinger equations and nonlinear groups . Funkcial. Ekvac. , 30 : 115 – 125 .
   Google Scholar
• Kato , T . 1989 . “ Nonlinear Schrödinger equations ” . In Schrödinger Operators, Lecture Notes in Physics , Edited by: Holden , H and Jensen , A . Vol. 345 , 218 – 263 . Berlin : Springer .
   Google Scholar
• Cazenave , T . 2003 . Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics , Vol. 10 , Providence : American Mathematical Society .
   Google Scholar
• Sulem , C and Sulem , P-L . 1999 . Nonlinear Schrödinger Equation, Applied Mathematical Sciences , Vol. 139 , New York : Springer-Verlag .
   Google Scholar
• Planchon , F , Stalker , J and Tahvildar-Zadeh , AS . 2003 . L p estimates for the wave equation with the inverse-square potential . Discr. Contin. Dyn. Syst. , 9 : 427 – 442 .
   Web of Science ®Google Scholar
• Vazquez , JL and Zuazua , E . 2000 . The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential . J. Funct. Anal. , 173 : 103 – 153 .
   Web of Science ®Google Scholar
• Tanabe , H . 1979 . Equations of Evolution, Monographs and Studies in Mathematics , Vol. 6 , London : Pitman .
   Google Scholar
• Cazenave , T and Haraux , A . 1998 . An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications , Vol. 13 , New York : Oxford University Press .
   Google Scholar
• Burq , N , Planchon , F , Stalker , J and Tahvildar-Zadeh , AS . 2003 . Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential . J. Funct. Anal. , 203 : 519 – 549 .
   Web of Science ®Google Scholar
• Okazawa , N and Yokota , T . 2010 . Subdifferential operator approach to strong wellposedness of the complex Ginzburg–Landau equation . Discrete Contin. Dynam. Systems , 28 : 311 – 341 .
   Web of Science ®Google Scholar
• Ogawa , T and Yokota , T . 2004 . Uniqueness and inviscid limits of solutions for the complex Ginzburg–Landau equation in a two-dimensional domain . Comm. Math. Phys. , 245 : 105 – 121 .
   Web of Science ®Google Scholar
• Bergh , J and Löfström , J . 1976 . Interpolation Spaces, An Introduction, Grundlehren der Mathematics Wissenschaften , Vol. 223 , Berlin and New York : Springer-Verlag .
   Google Scholar
• DiBenedetto , E . 2002 . “ Real Analysis ” . In Birkhäuser Advanced Texts: Basler Lehrbücher , Boston, Cambridge , MA : Birkhäuser .
   Google Scholar
• Christ , M and Kiselev , A . 2001 . Maximal functions associated to filtrations . J. Funct. Anal. , 179 : 409 – 425 .
   Web of Science ®Google Scholar
• Tao , T . Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, Vol. 106, AMS, Providence, RI, 2006
   Google Scholar
• Ozawa , T . 2006 . Remarks on proofs of conservation laws for nonlinear Schrödinger equations . Calc. Var. Partial Differ. Eqns , 25 : 403 – 408 .
   Web of Science ®Google Scholar
• Okazawa , N . 1996 . L p -theory of Schrödinger operators with strongly singular potentials . Japan. J. Math. , 22 : 199 – 239 .
   Google Scholar
• Shen , Z . 1995 . L p estimates for Schrödinger operators with certain potentials . Ann. Inst. Fourier, Grenoble , 45 : 513 – 546 .
   Web of Science ®Google Scholar
• Miyokawa , T and Shigekawa , I . 2006 . On the equivalence of L p -norms related to Schrödinger type operators on Riemannian manifolds . Probab. Theory Related Fields , 135 : 487 – 519 .
   Web of Science ®Google Scholar