WELL-POSEDNESS OF NONLINEAR FRACTIONAL
SCHRÖDINGER AND WAVE EQUATIONS
IN SOBOLEV SPACES

Abstract

In this paper, we establish the local well-posedness results in sub-critical and critical cases for the pure power-type nonlinear fractional Schrödinger and wave equations on $\R^d$, namely

\begin{displaymath}
i\partial_t u + \Lambda^\sigma u + \mu \vert u\vert^{\nu-1} u=0, \quad u_{\vert t=0} =\varphi,
\end{displaymath}

and

\begin{displaymath}
\partial^2_t v +\Lambda^{2\sigma} v + \mu \vert v\vert^{\nu...
...v_{\vert t=0}=\varphi, \quad \partial_t v_{\vert t=0} = \phi,
\end{displaymath}

where $\sigma \in (0,\infty)\backslash\{1\}, \nu>1, \mu \in \{\pm 1\}$ and $\Lambda =\sqrt{-\Delta}$ is the Fourier multiplier by $\vert\xi\vert$. For the nonlinear fractional Schrödinger equation, we extend the previous results in [#!HongSire!#] for $\sigma \geq 2$. These results cover the well-known results for Schrödinger equation $\sigma =2$ given in [#!CazenaveWeissler!#]. In the case $\sigma \in (0,2)\backslash\{1\}$, we show the local well-posedness in the sub-critical case for $\nu>1$ in contrast to $\nu\geq 2$ when $d=1$, and $\nu\geq 3$ when $d\geq 2$ of [#!HongSire!#]. These results also generalize the ones of [#!ChoHwangKwonLee!#] when $d=1$ and of [#!GuoHuo13!#] when $d\geq 2$, where the authors considered the cubic fractional Schrödinger equation with $\sigma \in (1,2)$. To our knowledge, the nonlinear fractional wave equation does not seem to have been much considered, up to [#!Wang!#] on the scattering operator with $\sigma$ an even integer and [#!ChenFanZhang14!#], [#!ChenFanZhang15!#] in the context of the damped fractional wave equation.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 31
Issue: 4
Year: 2018

DOI: 10.12732/ijam.v31i4.1

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] H. Bahouri, J. Y. Chemin, R. Danchin, Fourier Analysis and Non-linear Partial Differential Equations, Ser. of Comprehensive Studies in Mathematics 343, Springer (2011).
  2. [2] J. Bergh, J. Lofstom, Interpolation Spaces, Springer, New York (1976).
  3. [3] N. Burq, P. Gerard, N. Tzvetkov, Strichartz inequalities and the nonlinear Schr¨odinger equation on compact manifolds, Amer. J. Math., 126 (2004), 569-605.
  4. [4] T. Cazenave, F. B. Weissler, The Cauchy problem for the critical nonlinear Schr¨odinger equation in Hs, Nonlinear Anal., 14 (1990), 807-836.
  5. [5] T. Cazenave, Semilinear Schr¨odinger Equations, Courant Lecture Notes in Math. 10, Courant Institute of Mathematical Sciences, AMS (2003).
  6. [6] J. Chen, D. Fan, C. Zhang, Space-time estimates on damped fractional wave equations, Abstr. Appl. Anal., 2014 (2014), Art. ID 428909.
  7. [7] J. Chen, D. Fan, C. Zhang, Estimates for damped fractional wave equations and applications, Electron. J. Differ. Equ. Conf., 2015 (2015), No 162, 1- 14.
  8. [8] W. Chen, S. Holm, Physical interpretation of fractional diffusion-wave equation via lossy media obeying frequency power law, Physics Review, arXiv:math-ph/0303040 (2003).
  9. [9] C.H. Cho, Y. Koh, I. Seo, On inhomogeneous Strichartz estimates for fractional Schr¨odinger equations and theirs applications, Discrete Contin. Dyn. Syst., 36, No 4 (2016), 1905-1926.
  10. [10] Y. Cho, H. Hajaiej, G. Hwang, T. Ozawa, On the Cauchy problem of fractional Schr¨odinger equation with Hartree type nonlinearity, Funkcial. Ekvac., 56, No 2 (2013), 193-224.
  11. [11] Y. Cho, G. Hwang, S. Kwon, S. Lee, Well-posedness and ill-posedness for the cubic fractional Schr¨odinger equations, Discrete Contin. Dyn. Syst., 35, No 7 (2015), 2863-2880.
  12. [12] Y. Cho, T. Ozawa, S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10, No 4 (2011), 1121-1128.
  13. [13] M. Christ, I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100, No 1 (1991), 87-109.
  14. [14] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global wellposedness and scattering for the energy-critical nonlinear Schr¨odinger equation in R3, Ann. of Math., 167, No 3 (2008), 767-865.
  15. [15] J. Ginibre, G. Velo, On the global Cauchy problem for some nonlinear Schr¨odinger equations, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 2 (1984), 309-323.
  16. [16] J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Klein- Gordon equation, Math. Z., 189 (1985), 487-505.
  17. [17] B. Guo, Z. Huo, Global well-posedness for the fractional nonlinear Schr¨odinger equation, Comm. Partial Differential Equations 36, No 2 (2011), 247-255.
  18. [18] B. Guo, Z. Huo, Well-posedness for the nonlinear fractional Schr¨odinger equation and inviscid limit behavior of solution for the fractional Ginzburg- Landau equation, Fract. Calc. Appl. Anal., 16, No 1 (2013), 226-242; DOI: 10.2478/s13540-013-0014-y.
  19. [19] B. Guo, B. Wang, The global Cauchy problem and scattering of solutions for nonlinear Schr¨odinger equations in Hs, Differential Integral Equations 15, No 9 (2002), 1073-1083.
  20. [20] Z. Guo, Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schr¨odinger and wave equations, J. Anal. Math., 124, No 1 (2014), 1-38.
  21. [21] L. Grafakos, S. Oh, The Kato-Ponce inequality, Comm. Partial Differential Equations 39, No 6 (2014), 1128-1157.
  22. [22] Y. Hong, Y. Sire, On fractional Schr¨odinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14, No 6 (2015), 2265-2282.
  23. [23] A.D. Ionescu, F. Pusateri, Nonlinear fractional Schr¨odinger equations in one dimension, J. Func. Anal., 266 (2014), 139-176.
  24. [24] T. Kato, On nonlinear Schr¨odinger equations. II. Hs-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.
  25. [25] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120, No 5 (1998), 955-980.
  26. [26] N. Laskin, Fractional quantum mechanics and L´evy path integrals, Phys. Lett A 268 (2000), 298-305.
  27. [27] N. Laskin, Fractional Schr¨odinger equation, Phys. Rev. E 66 (2002), 056108.
  28. [28] H. Lindblad, C-D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130 (1995), 357-426.
  29. [29] C.H. Miao, Global strong solutions for nonlinear higher order Schr¨odinger equations, Acta. Math. Appl. Sinica 19 (1996), 211-221.
  30. [30] B. Pausader, Global well-posedness for energy critical fourth-order Schr¨odinger equations in the radial case, Dyn. Partial Differential Equations, 4, No 3 (2007), 197-225.
  31. [31] B. Pausader, Scattering and the Levandosky-Strauss conjecture for fourthorder nonlinear wave equations, J. Diff. Equa., 241 (2007), 237-278.
  32. [32] G. Staffilani, The Initial Value Problem for Some Dispersive Differential Equations, Dissertation, University of Chicago (1995).
  33. [33] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Series 30, Princeton University Press (1970).
  34. [34] T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics 106, AMS (2006).
  35. [35] M. Taylor, Tool for PDE Pseudodifferential Operators, Paradifferential Operators and Layer Potentials, Mathematical Surveys and Monographs 81, AMS (2000).
  36. [36] H. Triebel, Theory of Function Spaces, Birkh¨auser, Basel (1983).
  37. [37] B. Wang, Nonlinear scattering theory for a class of wave equations in Hs, J. Math. Anal. Appl., 296, No 1 (2004), 74-96.