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The Cauchy problem for the critical nonlinear Schrödinger equation in $$H^ s$$. (English) Zbl 0706.35127
Consider the Cauchy problem for the nonlinear Schrödinger equation in $${\mathbb{R}}^ N$$, $(NLS)\quad iu_ t+\Delta u+\lambda | u|^{\alpha}u=0,\quad u(0)=\phi.$ Here, $$\phi$$ is a given initial value, $$\lambda$$ is a real parameter and $$\alpha >0$$ is arbitrary (up to technical conditions that intervene only for dimensions $$N\geq 7)$$. It is established that if $$0\leq s<N/2$$ and $$\alpha\leq 4/(N-2s)$$, then problem (NLS) is locally well posed in the Sobolev space $$H^ s({\mathbb{R}}^ N)$$. In the case $$\alpha =4/(N-2s)$$, if $$\| (-\Delta)^{s/2}\phi \|_{L^ 2}$$ is sufficiently small, then the solution is global and decays as $$t\to \infty$$ at the same rate as the free solution. The method relies on estimates for $$e^{it\Delta}$$ in certain Besov spaces, on estimates of $$| u|^{\alpha}u$$ in the same Besov spaces, and on a fixed point argument.
Reviewer: Th.Cazenave

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35B40 Asymptotic behavior of solutions to PDEs
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##### References:
 [1] Bergh, J.; Löfstöm, J., Interpolation spaces, (1976), Springer New York [2] Cazenave, T.; Weissler, F.B., The Cauchy problem for the nonlinear Schrödinger equation in H1, Manuscripta math., 61, 477-494, (1988) · Zbl 0696.35153 [3] Cazenave, T.; Weissler, F.B., Some remarks on the nonlinear Schrödinger equation in the critical case, (), 18-29, Lecture Notes in Mathematics [4] Fabes, E.B.; Lewis, J.E.; Riviere, N.M., Boundary value problems for the navier – strokes equations, Am. J. math., 99, 626-668, (1977) · Zbl 0386.35037 [5] Fujita, H.; Kato, T., On the navier – strokes initial value problem I, Archs. ration mech. analysis, 16, 269-315, (1964) · Zbl 0126.42301 [6] Giga, Y., Solutions for semilinear parabolic equations in L^p and regularity of weak solutions for the Navier-Stokes system, J. diff. eqns, 62, 168-212, (1986) · Zbl 0577.35058 [7] Giga, Y.; Kohn, R.V., Characterizing blowup using similarity variables, Indiana univ. math. J., 36, 1-40, (1987) · Zbl 0601.35052 [8] Giga, Y.; Kohn, R.V.; Papanicolaou, G., Removability of blowup points for similinear heat equations, Proc. EQUADIFF conf., (August 1987) [9] Giga, Y.; Miyakawa, T., Solutions in L_r of the navier – stokes initial value problem, Archs. ration. mech. analysis, 89, 267-281, (1985) · Zbl 0587.35078 [10] Ginibre, J.; Velo, G., Sur une équation de Schrödinger non linéaire avec interaction non locate, (), 155-199, Collége de France Seminar · Zbl 0497.35024 [11] Ginibre, J.; Velo, G., The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. inst. H. Poincaré analyse non linéaire, 2, 309-327, (1985) · Zbl 0586.35042 [12] Ginibre, J.; Velo, G., The global Cauchy problem for the nonlinear klein – gordon equation, Math. Z., 189, 487-505, (1985) · Zbl 0549.35108 [13] Haraux, A.; Weissler, F.B., Non-uniqueness for a semilinear initial value problem, Indiana univ. math. J., 31, 167-189, (1982) · Zbl 0465.35049 [14] Kato, T., Strong L^p-solutions of the navier – stokes equations in \bfrm with applications to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073 [15] Kato, T., On nonlinear Schrödinger equations, Ann. inst. H. Poincaré physique théorique, 46, 113-129, (1987) · Zbl 0632.35038 [16] Kato, T.; Fujita, H., On the nonstationary navier – stokes system, Rc. semin. mat. univ. Padova, 32, 243-260, (1962) · Zbl 0114.05002 [17] Merle, F., Limit of the solution of the nonlinear schödinger equation at the blow-up time, J. funct. analysis, 84, 201-214, (1989) · Zbl 0681.35078 [18] Miyakawa, T., On the initial value problem for the navier – stokes equations in L^p spaces, Hiroshima math. J., 11, 9-20, (1981) · Zbl 0457.35073 [19] Mueller, C.E.; Weissler, F.B., Single point blow-up for a general semilinear heat equation, Indiana univ. math. J., 34, 881-913, (1985) · Zbl 0597.35057 [20] Reed, M., Abstract nonlinear wave equations, Lecture notes in mathematics, 507, (1976), Springer New York [21] Simon, J., Regularité de la composition de deux fonctions et applications, Boll. un. mat. ital., 16B, 5, 501-522, (1979) · Zbl 0409.35076 [22] Sobolevskii, P.E., Study of navier – stokes equations by the methods of the theory of parabolic equations in Banach spaces, Soviet math. dokl., 5, 720-723, (1964) [23] Stein, E.M., Singular integrals and differentiability properties of functions, (1970), Princeton University Press Princeton · Zbl 0207.13501 [24] Strauss, W.A., Nonlinear invariant wave equations, (), 197-249 [25] Strichartz, R., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke math. J., 44, 705-714, (1977) · Zbl 0372.35001 [26] Tsutsumi, Y., L2-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcialaj ekvacioj, 30, 115-125, (1987) · Zbl 0638.35021 [27] T\scSUTSUMI Y., Lower estimates of blow-up solutions for nonlinear Schrödinger equations, preprint. [28] von Wahl, W., Regularity questions for the navier – stokes equations, (), 538-542 [29] von Wahl, W., The equations of navier – stokes and abstract parabolic equations, (1985), Vieweg Wiesbaden · Zbl 1409.35168 [30] von Wahl, W., Global solutions to evolutions equations of parabolic type, (), 254-266 [31] Weinstein, M., On the structure and formation of singularities of solutions to nonlinear dispersive equations, Communs partial diff. eqns, 11, 545-565, (1986) · Zbl 0596.35022 [32] Weissler, F.B., Local existence and nonexistence for semilinear parabolic equations in L^p, Indiana univ. math J., 29, 79-102, (1980) · Zbl 0443.35034 [33] Weissler, F.B., The navier – stokes initial value problem in L^p, Archs ration. mech. analaysis, 74, 219-230, (1980) · Zbl 0454.35072 [34] Weissler, F.B., Existence and non-existence of global solutions for a semilinear heat equation, Israel J. math., 38, 29-40, (1981) · Zbl 0476.35043 [35] Weissler, F.B., L^p-energy and blow-up for a semilinear heat equation, (), 545-551, Part 2 [36] Yajima, K., Existence of solutions for Schrödinger evolutions equations, Communs math. phys., 110, 415-426, (1987) · Zbl 0638.35036
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