# American Institute of Mathematical Sciences

December  2012, 1(2): 337-354. doi: 10.3934/eect.2012.1.337

## Energy methods for abstract nonlinear Schrödinger equations

 1 Department of Mathematics, Science University of Tokyo, 1-3 Kagurazaka, Sinjuku-ku, Tokyo 162-8601 2 Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received  June 2012 Revised  July 2012 Published  October 2012

So far there seems to be no abstract formulations for nonlinear Schrödinger equations (NLS). In some sense Cazenave[2, Chapter 3] has given a guiding principle to replace the free Schrödinger group with the approximate identity of resolvents. In fact, he succeeded in separating the existence theory from the Strichartz estimates. This paper is a proposal to extend his guiding principle by using the square root of the resolvent. More precisely, the abstract theory here unifies the local existence of weak solutions to (NLS) with not only typical nonlinearities but also some critical cases. Moreover, the theory yields the improvement of [21].
Citation: Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337
##### References:
 [1] H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677.  doi: 10.1016/0362-546X(80)90068-1.  Google Scholar [2] T. Cazenave, "Semilinear Schrödinger Equations,'', Courant Lecture Notes in Mathematics, (2003).   Google Scholar [3] T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,'', Oxford Lecture Series in Mathematics and its Applications, (1998).   Google Scholar [4] T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$,, Manuscripta Math., 61 (1988), 477.  doi: 10.1007/BF01258601.  Google Scholar [5] T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case,, Nonlinear semigroups, (1987), 18.   Google Scholar [6] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,, Nonlinear Anal., 14 (1990), 807.  doi: 10.1016/0362-546X(90)90023-A.  Google Scholar [7] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case,, J. Funct. Anal., 32 (1979), 1.  doi: 10.1016/0022-1236(79)90076-4.  Google Scholar [8] J. Ginibre and G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 309.   Google Scholar [9] M. J. Goldberg, L. Vega and N. Visciglia, Couterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials,, Int. Math. Res. Not., (2006).   Google Scholar [10] H. Hoshino and Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations,, Funkcial. Ekvac., 34 (1991), 475.   Google Scholar [11] R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equation,, Nonlinear Anal., 15 (1990), 479.  doi: 10.1016/0362-546X(90)90128-4.  Google Scholar [12] T. Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 edition, (1980).   Google Scholar [13] T. Kato, On nonlinear Schrödinger equations,, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113.   Google Scholar [14] T. Kato, Nonlinear Schrödinger equations,, in, 345 (1989), 218.   Google Scholar [15] T. Kato, On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness,, J. Anal. Math., 67 (1995), 281.  doi: 10.1007/BF02787794.  Google Scholar [16] Y. Maeda and N. Okazawa, Holomorphic families of Schrödinger operators in $L^p$,, SUT J. Math., 47 (2011), 185.   Google Scholar [17] T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations,, Nonlinear Anal., 14 (1990), 765.  doi: 10.1016/0362-546X(90)90104-O.  Google Scholar [18] M. Ohta, Instability of bound states for abstract nonlinear Schrödinger equations,, J. Funct. Anal., 261 (2011), 90.  doi: 10.1016/j.jfa.2011.03.010.  Google Scholar [19] N. Okazawa, $L^p$-theory of Schrödinger operators with strongly singular potentials,, Japan. J. Math., 22 (1996), 199.   Google Scholar [20] N.\,Okazawa, Gauss hypergeometric functions of operators unifying fractional powers and logarithms,, Semigroups of Operators: Theory and Applications (Rio de Janeiro, (2001), 209.   Google Scholar [21] N. Okazawa, T. Suzuki and T. Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials,, Appl. Anal., 91 (2012), 1605.  doi: 10.1080/00036811.2011.631914).  Google Scholar [22] Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials,, Ann. Inst. Fourier, 45 (1995), 513.  doi: 10.5802/aif.1463.  Google Scholar [23] T. Suzuki, Energy methods for Hartree type equations with inverse-square potentials,, preprint., ().   Google Scholar [24] H. Tanabe, "Equations of Evolution,'', Monographs and Studies in Mathematics vol. 6, (1979).   Google Scholar [25] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups,, Funkcial. Ekvac., 30 (1987), 115.   Google Scholar [26] Y. Tsutsumi, Global strong solutions for nonlinear Schrödinger equations,, Nonlinear Anal., 11 (1987), 1143.  doi: 10.1016/0362-546X(87)90003-4.  Google Scholar [27] F. B. Weissler, Local existence and nonexistence for semilinear parabolic equation in $L^p$,, Indiana Univ. Math. J., 29 (1980), 79.  doi: 10.1512/iumj.1980.29.29007.  Google Scholar

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##### References:
 [1] H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677.  doi: 10.1016/0362-546X(80)90068-1.  Google Scholar [2] T. Cazenave, "Semilinear Schrödinger Equations,'', Courant Lecture Notes in Mathematics, (2003).   Google Scholar [3] T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,'', Oxford Lecture Series in Mathematics and its Applications, (1998).   Google Scholar [4] T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$,, Manuscripta Math., 61 (1988), 477.  doi: 10.1007/BF01258601.  Google Scholar [5] T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case,, Nonlinear semigroups, (1987), 18.   Google Scholar [6] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,, Nonlinear Anal., 14 (1990), 807.  doi: 10.1016/0362-546X(90)90023-A.  Google Scholar [7] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case,, J. Funct. Anal., 32 (1979), 1.  doi: 10.1016/0022-1236(79)90076-4.  Google Scholar [8] J. Ginibre and G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 309.   Google Scholar [9] M. J. Goldberg, L. Vega and N. Visciglia, Couterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials,, Int. Math. Res. Not., (2006).   Google Scholar [10] H. Hoshino and Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations,, Funkcial. Ekvac., 34 (1991), 475.   Google Scholar [11] R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equation,, Nonlinear Anal., 15 (1990), 479.  doi: 10.1016/0362-546X(90)90128-4.  Google Scholar [12] T. Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 edition, (1980).   Google Scholar [13] T. Kato, On nonlinear Schrödinger equations,, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113.   Google Scholar [14] T. Kato, Nonlinear Schrödinger equations,, in, 345 (1989), 218.   Google Scholar [15] T. Kato, On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness,, J. Anal. Math., 67 (1995), 281.  doi: 10.1007/BF02787794.  Google Scholar [16] Y. Maeda and N. Okazawa, Holomorphic families of Schrödinger operators in $L^p$,, SUT J. Math., 47 (2011), 185.   Google Scholar [17] T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations,, Nonlinear Anal., 14 (1990), 765.  doi: 10.1016/0362-546X(90)90104-O.  Google Scholar [18] M. Ohta, Instability of bound states for abstract nonlinear Schrödinger equations,, J. Funct. Anal., 261 (2011), 90.  doi: 10.1016/j.jfa.2011.03.010.  Google Scholar [19] N. Okazawa, $L^p$-theory of Schrödinger operators with strongly singular potentials,, Japan. J. Math., 22 (1996), 199.   Google Scholar [20] N.\,Okazawa, Gauss hypergeometric functions of operators unifying fractional powers and logarithms,, Semigroups of Operators: Theory and Applications (Rio de Janeiro, (2001), 209.   Google Scholar [21] N. Okazawa, T. Suzuki and T. Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials,, Appl. Anal., 91 (2012), 1605.  doi: 10.1080/00036811.2011.631914).  Google Scholar [22] Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials,, Ann. Inst. Fourier, 45 (1995), 513.  doi: 10.5802/aif.1463.  Google Scholar [23] T. Suzuki, Energy methods for Hartree type equations with inverse-square potentials,, preprint., ().   Google Scholar [24] H. Tanabe, "Equations of Evolution,'', Monographs and Studies in Mathematics vol. 6, (1979).   Google Scholar [25] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups,, Funkcial. Ekvac., 30 (1987), 115.   Google Scholar [26] Y. Tsutsumi, Global strong solutions for nonlinear Schrödinger equations,, Nonlinear Anal., 11 (1987), 1143.  doi: 10.1016/0362-546X(87)90003-4.  Google Scholar [27] F. B. Weissler, Local existence and nonexistence for semilinear parabolic equation in $L^p$,, Indiana Univ. Math. J., 29 (1980), 79.  doi: 10.1512/iumj.1980.29.29007.  Google Scholar
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