Concentration for blow-up solutions of semi-relativistic Hartree equations of critical type. (English) Zbl 1524.35585
Summary: This paper concerns the semi-relativistic Hartree equation
\[
{i} \partial_t u = \sqrt{- \Delta + m^2} u -(| \cdot |^{- 1} \ast | u |^2) u
\]
in \(\mathbb{R}^3\). We prove the concentration results for finite time blow-up solutions with general \(H_x^{1 / 2}(\mathbb{R}^3)\) data, and show the relation between the concentration rate and the blow-up order.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q40 | PDEs in connection with quantum mechanics |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 85A15 | Galactic and stellar structure |
References:
| [1] | Elgart, A.; Schlein, B., Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60, 500-545, (2007) · Zbl 1113.81032 |
| [2] | Cho, Y.; Ozawa, T., On radial solutions of semi-relativistic Hartree equations, Discrete Contin. Dyn. Syst. Ser. S, 1, 71-82, (2008) · Zbl 1147.35077 |
| [3] | Herr, S.; Tesfahun, A., Small data scattering for semi-relativistic equations with Hartree type nonlinearity, J. Differential Equations, 259, 5510-5532, (2015) · Zbl 1321.35178 |
| [4] | Lenzmann, E., Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10, 43-64, (2007) · Zbl 1171.35474 |
| [5] | Fröhlich, J.; Jonsson, B. L.G.; Lenzmann, E., Boson stars as solitary waves, Comm. Math. Phys., 274, 1-30, (2007) · Zbl 1126.35064 |
| [6] | Fröhlich, J.; Lenzmann, E., Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60, 1691-1705, (2007) · Zbl 1135.35011 |
| [7] | Hmidi, T.; Keraani, S., Blowup theory for the critical nonlinear Schrödinger equations revisited, Internat. Math. Res. Notices, 46, 2815-2828, (2005) · Zbl 1126.35067 |
| [8] | Keraani, S., On the blow-up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235, 171-192, (2006) · Zbl 1099.35132 |
| [9] | Merle, F.; Tsutsumi, Y., \(L^2\)-concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations, 84, 205-214, (1990) · Zbl 0722.35047 |
| [10] | Fröhlich, J.; Lenzmann, E., Dynamical collapse of white dwarfs in Hartree and Hartree-Fock theorey, Comm. Math. Phys., 274, 737-750, (2007) · Zbl 1130.85004 |
| [11] | Miao, C.; Xu, G.; Zhao, L., Global well-posedness and scattering for the mass-critical Hartree equation for radial data, J. Math. Pures Appl., 91, 49-79, (2009) · Zbl 1154.35078 |
| [12] | Brézis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88, 486-490, (1983) · Zbl 0526.46037 |
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