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Selfsimilar expanders of the harmonic map flow. (English) Zbl 1246.35059

The authors study the existence, uniqueness and stability of self-similar expanders of the harmonic map heat flow in equivariant settings. They prove the existence of self-similar solutions to any admissible initial data and that the stability properties are essentially determined by the energy minimizing properties of the equator maps.

MSC:

35C06 Self-similar solutions to PDEs
35B35 Stability in context of PDEs
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[1] Bertsch, M.; Dal Passo, R.; Van der Hout, R., Nonuniqueness for the heat flow of harmonic maps on the disk, Arch. Rat. Mech. Anal., 161, 2, 93-112 (2002) · Zbl 1006.35050
[2] P. Biernat, P. Bizon, private communication.; P. Biernat, P. Bizon, private communication.
[3] Brezis, H.; Coron, J. M.; Lieb, E. H., Harmonic maps with defects, Comm. Math. Phys., 107, 649-705 (1986) · Zbl 0608.58016
[4] Cazenave, T.; Shatah, J.; Tahvildar-Zadeh, S., Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields, Ann. Inst. H. Poincaré Phys. Théor., 68, 3, 315-349 (1998) · Zbl 0918.58074
[5] Chen, Y., The weak solutions to the evolution problems of harmonic maps, Math. Z., 201, 1, 69-74 (1989) · Zbl 0685.58015
[6] Chen, Y.; Struwe, M., Existence and partial regularity results for the heat flow for harmonic maps, J. Differential Geometry, 201, 1, 83-103 (1989) · Zbl 0652.58024
[7] Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations, International Series in Pure and Applied Mathematics (1955), McGraw-Hill · Zbl 0042.32602
[8] Coron, J.-M., Nonuniqueness for the heat flow of harmonic maps, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 7, 4, 335-344 (1990) · Zbl 0707.58017
[9] Eells, J.; Ratto, A., Harmonic Maps and Minimal Immersions with Symmetries, Annals of Mathematics Studies (1993), Princeton University Press · Zbl 0783.58003
[10] Escobedo, M.; Kavian, O., Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal., 11, 10, 1103-1133 (1987) · Zbl 0639.35038
[11] Fan, H., Existence of the self-similar solutions in the heat flow of harmonic maps, Sci. China, Ser. A, 42, 113-132 (1999) · Zbl 0926.35021
[12] Freire, A., Uniqueness of the harmonic map flow in two dimensions, Calc. Var., 3, 1, 95-105 (1995) · Zbl 0814.35057
[13] Galaktionov, V.; Vazquez, J., Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math., 50, 1, 1-67 (1997) · Zbl 0874.35057
[14] Gastel, A., Regularity theory for minimizing equivariant (p-)harmonic mappings, Calc. Var., 6, 4, 329-367 (1998) · Zbl 0911.58010
[15] Gastel, A., The extrinsic polyharmonic map heat flow in the critical dimension, Adv. Geom., 6, 10, 595-613 (2006) · Zbl 1136.58010
[16] Germain, P., On the existence of smooth self-similar blowup profiles for the wave map equation, Comm. Pure Appl. Math., 62, 5, 706-728 (2009) · Zbl 1179.35033
[17] Haraux, A.; Weissler, F., Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31, 2, 167-189 (1982) · Zbl 0465.35049
[18] Hong, M.-C., Some new examples for nonuniqueness of the evolution problem of harmonic maps, Comm. Anal. Geom., 6, 4, 809-818 (1998) · Zbl 0949.58016
[19] T. Ilmanen, Lectures on mean curvature flow and related equations, Lecture Notes ICTP, Trieste, 1995.; T. Ilmanen, Lectures on mean curvature flow and related equations, Lecture Notes ICTP, Trieste, 1995.
[20] Kaul, H.; Jäger, W., Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems, J. Reine Angew. Math., 343, 146-161 (1983) · Zbl 0516.35032
[21] Koch, H.; Lamm, T., Geometric flows with rough initial data (2009)
[22] Lin, F.; Wang, C., On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chinese Ann. Math. Ser. B, 31, 6, 921-938 (2010) · Zbl 1208.35002
[23] Lin, F.; Wang, C., The Analysis of Harmonic Maps and Their Heat Flows (2008), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ
[24] Naito, Y., Non-uniqueness of solutions to the Cauchy problem for semilinear heat equations with singular initial data, Math. Ann., 329, 1, 161-196 (2004) · Zbl 1059.35055
[25] Naito, Y., An ode approach to the multiplicity of self-similar solutions for semi-linear heat equations, Proc. Roy. Soc. Edinburgh Sect. A, 4, 807-835 (2006) · Zbl 1112.35100
[26] Peletier, L.; Terman, D.; Weissler, F., On the equation \(\delta u + \frac{1}{2} x \Delta \nabla u + f(u) = 0\), Arch. Rat. Mech. Anal., 94, 1, 83-99 (1986) · Zbl 0615.35034
[27] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1984), Springer-Verlag · Zbl 0153.13602
[28] Quittner, P.; Souplet, P., Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts Basler Lehrbücher (2007), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1128.35003
[29] Reed, M.; Simon, B., Methods of Modern Mathematical Physics, vol. 4: Analysis of Operators (1972), Academic Press
[30] Rubinstein, J.; Sternberg, P.; Keller, J., Reaction-diffusion processes and evolution to harmonic maps, SIAM J. Appl. Math., 49, 6, 1722-1733 (1995) · Zbl 0702.35128
[31] Rupflin, M., An improved uniqueness result for the harmonic map flow in two dimensions, Calc. Var., 33, 3, 329-341 (2008) · Zbl 1157.58004
[32] M. Rupflin, Harmonic map flow and variants, PhD thesis, ETH Zurich, 2010.; M. Rupflin, Harmonic map flow and variants, PhD thesis, ETH Zurich, 2010.
[33] Shatah, J., Weak solutions and development of singularities of the \(su(2)σ\)-model, Comm. Pure Appl. Math., 41, 4, 459-469 (1988) · Zbl 0686.35081
[34] Showalter, R., Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, vol. 49 (1997), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0870.35004
[35] Souplet, P.; Weissler, F., Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20, 2, 213-235 (2003) · Zbl 1029.35106
[36] Struwe, M., On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv., 60, 558-581 (1985) · Zbl 0595.58013
[37] Struwe, M., On the evolution of harmonic maps in higher dimensions, J. Differential Geometry, 28, 3, 485-502 (1988) · Zbl 0631.58004
[38] Topping, P., Reverse bubbling and nonuniqueness in the harmonic map flow, Int. Math. Research Notices, 10, 558-581 (2002) · Zbl 1003.58014
[39] Vazquez, J. L.; Zuazua, E., The hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173, 103-153 (2000) · Zbl 0953.35053
[40] Wang, C.-Y., Bubble phenomena of certain palais-smale sequences from surfaces to general targets, Houston J. Math., 22, 559-590 (1996) · Zbl 0879.58019
[41] Weissler, F., Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations, Arch. Rat. Mech. Anal., 91, 3, 247-266 (1985) · Zbl 0604.34034
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