Skip to main content
SpringerLink
Log in

Dynamics Near the Ground State for the Energy Critical Nonlinear Heat Equation in Large Dimensions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the energy critical semilinear heat equation

$$\partial_tu = \Delta u + |u|^{\frac{4}{d-2}}u, \quad x \in \mathbb{R}^d$$

and give a complete classification of the flow near the ground state solitary wave

$$Q(x) = \frac{1}{\left(1+\frac{|x|^2}{d(d-2)}\right)^{\frac{d-2}{2}}}$$

in dimension \({d \ge 7}\), in the energy critical topology and without radial symmetry assumption. Given an initial data \({Q + \varepsilon_0}\) with \({\|\nabla \varepsilon_0\|_{L^2} \ll 1}\), the solution either blows up in the ODE type I regime, or dissipates, and these two open sets are separated by a codimension one set of solutions asymptotically attracted by the solitary wave. In particular, non self similar type II blow up is ruled out in dimension \({d \ge 7}\) near the solitary wave even though it is known to occur in smaller dimensions (Schweyer, J Funct Anal 263(12):3922–3983, 2012). Our proof is based on sole energy estimates deeply connected to Martel et al. (Acta Math 212(1):59–140, 2014) and draws a route map for the classification of the flow near the solitary wave in the energy critical setting. A by-product of our method is the classification of minimal elements around Q belonging to the unstable manifold.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin T.: Problemes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11(4), 573–598 (1976)

    MATH  Google Scholar 

  2. Bahouri H., Chemin J.Y., Danchin R.: Fourier analysis and nonlinear partial differential equations. (Vol. 343). Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  3. Brezis H., Cazenave T.: A nonlinear heat equation with singular initial data. J. D’Analyse Math. 68(1), 277–304 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Collot, C.: Type II blow up manifolds for the energy supercritical wave equation (2014). arXiv:1407.4525

  5. Collot, C.: Non radial type II blow up for the energy supercritical semilinear heat equation. (2016, preprint)

  6. Collot, C, Merle, F., Raphaël, P.: Stability of type I blow up for the energy critical heat equation. C. R. Math. Acad. Sci. Paris (to appear)

  7. Collot, C., Merle, F., Raphaël, P.: Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions (2016). arXiv:1604.08323

  8. Duyckaerts, T., Merle, F.: Dynamics of threshold solutions for energy-critical wave equation. Int. Math. Res. Pap. IMRP 2007, Art. ID rpn002, p. 67 (2008)

  9. Duyckaerts T., Merle F.: Dynamic of threshold solutions for energy-critical NLS. Geom. Funct. Anal. 18, 1787–1840 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Duyckaerts T., Kenig C., Merle F.: Classification of radial solutions of the focusing, energy-critical wave equation. Camb. J. Math. 1(1), 75–144 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fermanian Kammerer C., Merle F., Zaag H.: Stability of the blow-up profile of non-linear heat equations from the dynamical system point of view. Math. Ann. 317(2), 347–387 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Filippas, S., Herrero, M. A., Velazquez, J.J.: Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 456(2004), 2957–2982 (2000)

  13. Gidas B., Ni W.M., Nirenberg L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209–243 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Giga Y.: On elliptic equations related to self-similar solutions for nonlinear heat equations. Hiroshima Math. J. 16(3), 539–552 (1986)

    MathSciNet  MATH  Google Scholar 

  15. Giga Y., Kohn R.V.: Asymptotically self-similar blow-up of semilinear heat equations. Commun. Pure Appl. Math. 38(3), 297–319 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  16. Giga Y., Kohn R.V.: Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36, 1–40 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Giga Y., Kohn R.V.: Nondegeneracy of blowup for semilinear heat equations. Commun. Pure Appl. Math. 42(6), 845–884 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Giga, Y., Matsui, S.Y., Sasayama, S.: Blow up rate for semilinear heat equations with subcritical nonlinearity. Indiana Univ. Math. J. 53(2), 483–514 (2004)

  19. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der mathematischen Wissenschaften 224. Springer, Berlin (1998)

  20. Ladyzhenskaia, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and quasi-linear equations of parabolic type, AMS Trans. 23, Providence, RI (1968)

  21. Gustafson S., Nakanishi K., Tsai T.P.: Asymptotic stability, concentration and oscillations in harmonic map heat flow, Landau Lifschitz and Schrödinger maps on \({\mathbb{R}^2}\). Commun. Math. Phys. 300, 205–242 (2010)

    Article  ADS  MATH  Google Scholar 

  22. Herrero M.A., Velázquez J.J.L.: Explosion de solutions des èquations paraboliques semilinéaires supercritiques. C. R. Acad. Sci. Paris 319, 141–145 (1994)

    MathSciNet  Google Scholar 

  23. Krieger J., Schlag W., Tataru D.: Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math. 171, 543–615 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Krieger J., Schlag W., Tataru D.: Slow blow-up solutions for the \({H^1(\mathbb{R}^3)}\) critical focusing semilinear wave equation. Duke Math. J. 147, 1–53 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Martel Y., Merle F., Raphaël P.: Blow up for the critical generalized Korteweg-de Vries equation. I: Dynamics near the soliton. Acta Math. 212(1), 59–140 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Martel Y., Merle F., Raphaël P.: Blow up for the critical gKdV equation. II: minimal mass dynamics. J. Eur. Math. Soc. (JEMS) 17(8), 1855–1925 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Martel Y., Merle F., Nakanishi K., Raphaël P.: Codimension one threshold manifold for the critical gKdV equation. Commun. Math. Phys. 342(3), 1075–1106 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Matano H., Merle F.: Classification of type I and type II behaviors for a supercritical nonlinear heat equation. J. Funct. Anal. 256(4), 992–1064 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Matano H., Merle F.: On nonexistence of type II blowup for a supercritical nonlinear heat equation. Commun. Pure Appl. Math. 57(11), 1494–1541 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Matano, H., Merle, F.: Asymptotic soliton resolution for solutions of the critical nonlinear heat equation (2016, in preparation)

  31. Merle F.: On uniqueness and continuation properties after blow up time of self similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass. Commun. Pure Appl. Math. 45(2), 203–254 (1992)

    Article  MATH  Google Scholar 

  32. Merle F., Raphaël P.: Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation. J. Am. Math. Soc. 19(1), 37–90 (2006)

    Article  MATH  Google Scholar 

  33. Merle F., Raphaël P.: Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation. Commun. Math. Phys. 253, 675–704 (2005)

    Article  ADS  MATH  Google Scholar 

  34. Merle F., Raphaël P.: Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation. Ann. Math. 161(1), 157–222 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  35. Merle F., Raphaël P., Rodnianski I.: Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map. Invent. Math. 193(2), 249–365 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Merle, F., Raphaël, P., Rodnianski, I.: Type II blow up for the energy super critical NLS. Camb. Math. J. arXiv:1407.1415

  37. Merle F., Raphaël P., Szeftel J.: The instability of Bourgain-Wang solutions for the \({L^2}\) critical NLS. Am. J. Math. 135(4), 967–1017 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Merle F., Zaag H.: Optimal estimates for blowup rate and behavior for nonlinear heat equations. Commun. Pure Appl. Math. 51(2), 139–196 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  39. Merle F., Zaag H.: A Liouville theorem for vector-valued nonlinear heat equations and applications. Math. Ann. 316(1), 103–137 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mizoguchi N.: Type-II blowup for a semilinear heat equation. Adv. Differ. Equ. 9(11-12), 1279–1316 (2004)

    MathSciNet  MATH  Google Scholar 

  41. Nakanishi K., Schlag W.: Global dynamics above the ground state energy for the cubic NLS equation in 3D. Calc. Var. Partial Differ. Equ. 44(1-2), 1–45 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Raphaël P., Rodnianski I.: Stable blow up dynamics for critical corotational wave maps and the equivariant Yang Mills problems. Publ. Math. Inst. Hautes Etudes Sci. 115, 1–122 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Raphaël P., Schweyer R.: Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow. Comm. Pure Appl. Math. 66(3), 414–480 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. Raphaël P., Schweyer R.: Quantized slow blow up dynamics for the corotational energy critical harmonic heat flow. Anal. PDE 7(8), 1713–1805 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Raphaël P., Szeftel J.: Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS. J. Am. Math. Soc. 24(2), 471–546 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Reed S., Simon B.: Methods of modern mathematical physics, I-IV, Functional analysis, Second edition. Academic Press, Inc., New York (1980)

    MATH  Google Scholar 

  47. Schlag, W.: Spectral theory and nonlinear PDE: a survey (2005). arXiv:math/0509019

  48. Schweyer R.: Type II blow-up for the four dimensional energy critical semi linear heat equation. J. Funct. Anal. 263(12), 3922–3983 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  49. Talenti G.: Best constant in Sobolev inequality. Annali di Matematica pura ed Applicata 110(1), 353–372 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  50. Weissler F.B.: Local existence and nonexistence for semilinear parabolic equations in Lp. Indiana Univ. Math. J 29(1), 79–102 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire J.A. Dieudonné, Université de Nice-Sophia Antipolis, Nice, France

    Charles Collot & Pierre Raphaël

  2. Laboratoire Laga, Université de Cergy Pontoise, Cergy Pontoise, France

    Frank Merle

  3. IHES, Bures-sur-Yvette, France

    Frank Merle

Authors
  1. Charles Collot
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frank Merle
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pierre Raphaël
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre Raphaël.

Additional information

Communicated by W. Schlag

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Collot, C., Merle, F. & Raphaël, P. Dynamics Near the Ground State for the Energy Critical Nonlinear Heat Equation in Large Dimensions. Commun. Math. Phys. 352, 215–285 (2017). https://doi.org/10.1007/s00220-016-2795-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-016-2795-4

Search

Navigation