# American Institute of Mathematical Sciences

May  2013, 12(3): 1307-1319. doi: 10.3934/cpaa.2013.12.1307

## The regularity for a class of singular differential equations

 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received  February 2012 Revised  May 2012 Published  September 2012

We find an iteration technique and thus prove the optimal global regularity for the boundary value problem of a class of singular differential equations with strongly singular lower terms at the boundary. As applications, we obtain the regularity for the radial solutions of Ginzburg-Landau equations and harmonic maps.
Citation: Huaiyu Jian, Xiaolin Liu, Hongjie Ju. The regularity for a class of singular differential equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1307-1319. doi: 10.3934/cpaa.2013.12.1307
##### References:
 [1] F. Bethuel, H. Brezis and F. Helein, "Ginzburg-Landau Vortices,", Birkhauser, (1993).  doi: 10.1007/978-1-4612-0287-5.  Google Scholar [2] K. Q. Chang, W. Y. Ding and R. G. Ye, Finite time blow-up of the heat flow of harmonic maps from spheres,, J. Differential Geom., 36 (1992), 507.   Google Scholar [3] T. R. Ding and C. Z. Li, "A Course on Ordinary Diferential Equations,", Higher Educational Press, (1991).   Google Scholar [4] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Seconder Order,", Springer-Verlag, (2001).  doi: 10.1007/978-3-642-61798-0.  Google Scholar [5] M. Guan, S. Gustafson and T. P. Tsai, Global existence and blow-up for harmonic map heat flow,, J. Differential Equations, 246 (2009), 1.  doi: 10.1016/j.jde.2008.09.011.  Google Scholar [6] C. F. Gui, H. Y. Jian and H. J. Ju, Properties of translating solutions to mean curvature flow,, Discrete Contin. Dyn. Syst., 28 (2010), 441.  doi: 10.3934/dcds.2010.28.441.  Google Scholar [7] R. M. Herve and M. Herve, Qualitative study of the real solutions of a differential equation associated with the Ginzburg-Landau equation,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 11 (1994), 427.   Google Scholar [8] H. Y. Jian, H. J. Ju and W. Sun, Traveling fronts of curve flow with external force field,, Commun. Pure Appl. Anal., 9 (2010), 975.  doi: 10.3934/cpaa.2010.9.975.  Google Scholar [9] H. Y. Jian and Y. N. Liu, Ginzburg-Landau vortex and mean curvature flow with external force field,, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1831.  doi: 10.1007/s10114-005-0698-y.  Google Scholar [10] H. Y. Jian and B. Song, Vortex dynamics of Ginzburg-Landau equations in inhomogeneous superconductors,, J. Differential Equations, 170 (2001), 123.  doi: 10.1006/jdeq.2000.3822.  Google Scholar [11] H. Y. Jian and X. J. Wang, Bernsterin theorem and regularity for a class of Monge Amp\ere equations,, preprint, (2010).   Google Scholar [12] H. Y. Jian and X. J. Wang, Global regularity for fully nonlinear singular elliptic equations,, preprint, (2011).   Google Scholar [13] H. Y. Jian and X. W. Xu, The vortex dynamics of a Ginzburg-Landau system under pinning effect,, Science in China Ser. A, 46 (2003), 488.  doi: 10.1007/BF02884020.  Google Scholar [14] H. J. Ju, J. Lu and H. Y. Jian, Translating solutions to mean curvature flow with a forcing term in Minkowski space,, Commun. Pure Appl. Anal., 9 (2010), 963.  doi: 10.3934/cpaa.2010.9.963.  Google Scholar [15] P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equations,, J. Funct. Anal., 130 (1995), 334.  doi: 10.1006/jfan.1995.1073.  Google Scholar [16] P. Raphael and R. Schweyer, Stable blow-up dynamics for 1-corotational enenrgy critical harmonic heat flow,, preprint, ().   Google Scholar

show all references

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