Mathematische Annalen

, Volume 357, Issue 1, pp 89–163 | Cite as

Nonscattering solutions and blowup at infinity for the critical wave equation

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Abstract

We consider the critical focusing wave equation \((-\partial _t^2+\Delta )u+u^5=0\) in \({\mathbb{R }}^{1+3}\) and prove the existence of energy class solutions which are of the form
$$\begin{aligned} u(t,x)=t^\frac{\mu }{2}W(t^\mu x)+\eta (t,x) \end{aligned}$$
in the forward lightcone \(\{(t,x)\in {\mathbb{R }}\times {\mathbb{R }}^3: |x|\le t, t\gg 1\}\) where \(W(x)=(1+\frac{1}{3} |x|^2)^{-\frac{1}{2}}\) is the ground state soliton, \(\mu \) is an arbitrary prescribed real number (positive or negative) with \(|\mu |\ll 1\), and the error \(\eta \) satisfies
$$\begin{aligned} \Vert \partial _t \eta (t,\cdot )\Vert _{L^2(B_t)} +\Vert \nabla \eta (t,\cdot )\Vert _{L^2(B_t)}\ll 1,\quad B_t:=\{x\in {\mathbb{R }}^3: |x|<t\} \end{aligned}$$
for all \(t\gg 1\). Furthermore, the kinetic energy of \(u\) outside the cone is small. Consequently, depending on the sign of \(\mu \), we obtain two new types of solutions which either concentrate as \(t\rightarrow \infty \) (with a continuum of rates) or stay bounded but do not scatter. In particular, these solutions contradict a strong version of the soliton resolution conjecture.
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References

  1. 1.
    Bahouri, H., Gérard, P.: High frequency approximation of solutions to critical nonlinear wave equations. Am. J. Math. 121(1), 131–175 (1999)MATHCrossRefGoogle Scholar
  2. 2.
    Bahouri, H., Shatah, J.: Decay estimates for the critical semilinear wave equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 15(6), 783–789 (1998)Google Scholar
  3. 3.
    Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Springer, Berlin (1976, Grundlehren der Mathematischen Wissenschaften, No. 223)Google Scholar
  4. 4.
    Piotr, B., Tadeusz, C., Zbisław, T.: On blowup for semilinear wave equations with a focusing nonlinearity. Nonlinearity 17(6), 2187–2201 (2004)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Deift, P., Trubowitz, E.: Inverse scattering on the line. Comm. Pure Appl. Math. 32(2), 121–251 (1979)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Donninger, R., Schörkhuber, B.: Stable blow up dynamics for energy supercritical wave equations. Preprint arXiv:1207.7046 (2012)Google Scholar
  7. 7.
    Donninger, R., Schörkhuber, B.: Stable self-similar blow up for energy subcritical wave equations. Dyn. Partial Diff. Equ. 9(1), 63–87 (2012)MATHGoogle Scholar
  8. 8.
    Dunford, N., Schwartz, J.T.: Linear operators. Spectral theory. Self adjoint operators in Hilbert space, Part II. With the assistance of William G. Bade and Robert G. Bartle. Wiley, New York (1963)Google Scholar
  9. 9.
    Duyckaerts, T., Kenig, C., Merle, F.: Universality of the blow-up profile for small type II blow-up solutions of energy-critical wave equation: the non-radial case. Preprint arXiv:1003.0625 (2010)Google Scholar
  10. 10.
    Duyckaerts, T., Kenig, C., Merle, F.: Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation. J. Eur. Math. Soc. (JEMS) 13(3), 533–599 (2011)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Duyckaerts, T., Kenig, C., Merle, F.: Classification of radial solutions of the focusing, energy-critical wave equation. Preprint arXiv:1204.0031 (2012)Google Scholar
  12. 12.
    Duyckaerts, T., Kenig, C., Merle, F.: Profiles of bounded radial solutions of the focusing, energy-critical wave equation. Geom. Funct. Anal. (2012, to appear)Google Scholar
  13. 13.
    Duyckaerts, T., Merle, F.: Dynamics of threshold solutions for energy-critical wave equation. Int. Math. Res. Pap. IMRP, pages Art ID rpn002, 67, (2008)Google Scholar
  14. 14.
    Gesztesy, F., Maxim, Z.: On spectral theory for Schrödinger operators with strongly singular potentials. Math. Nachr. 279(9–10), 1041–1082 (2006)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34(4), 525–598 (1981)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ginibre, J., Soffer, A., Velo, G.: The global Cauchy problem for the critical nonlinear wave equation. J. Funct. Anal. 110(1), 96–130 (1992)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Grillakis, M.G.: Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity. Ann. Math. (2) 132(3), 485–509 (1990)Google Scholar
  18. 18.
    Hillairet, M., Raphaël, P.: Smooth type II blow up solutions to the four dimensional energy critical wave equation. Preprint arXiv:1010.1768 (2010)Google Scholar
  19. 19.
    Johnson, R.A., Bin, P.X., Yingfei, Y.: Positive solutions of super-critical elliptic equations and asymptotics. Comm. Partial Diff. Equ. 18(5–6), 977–1019 (1993)MATHCrossRefGoogle Scholar
  20. 20.
    Lev, K.: Global and unique weak solutions of nonlinear wave equations. Math. Res. Lett. 1(2), 211–223 (1994)MathSciNetMATHGoogle Scholar
  21. 21.
    Kenig, C.E., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201(2), 147–212 (2008)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Krieger, J., Schlag, W.: On the focusing critical semi-linear wave equation. Am. J. Math. 129(3), 843–913 (2007)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Krieger, J., Schlag, W., Tataru, D.: Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math. 171(3), 543–615 (2008)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Krieger, J., Schlag, W., Tataru, D.: Renormalization and blow up for the critical Yang-Mills problem. Adv. Math. 221(5), 1445–1521 (2009)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Krieger, J., Nakanishi, K., Schlag, W.: Global dynamics away from the ground state for the energy-critical nonlinear wave equation. Preprint arXiv:1010.3799 (2010)Google Scholar
  26. 26.
    Krieger, J., Nakanishi, K., Schlag, W.: Global dynamics of the nonradial energy-critical wave equation above the ground state energy. Preprint arXiv:1112.5663 (2011)Google Scholar
  27. 27.
    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), 1–53 (2009)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Levine, H.A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form \(Pu_{tt}=-Au+{\cal \,F}(u)\). Trans. Am. Math. Soc. 192, 1–21 (1974)MATHGoogle Scholar
  29. 29.
    Lindblad, H., Sogge, C.D.: On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal. 130(2), 357–426 (1995)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Merle, F., Raphael, P.: On universality of blow-up profile for \(L^2\) critical nonlinear Schrödinger equation. Invent. Math. 156(3), 565–672 (2004)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Merle, F., Raphael, P.: The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. Math. (2) 161(1):157–222 (2005)Google Scholar
  32. 32.
    Merle, F., Raphael, P.: On a sharp lower bound on the blow-up rate for the \(L^2\) critical nonlinear Schrödinger equation. J. Am. Math. Soc. 19(1):37–90 (2006, electronic)Google Scholar
  33. 33.
    Merle, F., Zaag, H.: Determination of the blow-up rate for the semilinear wave equation. Am. J. Math. 125(5), 1147–1164 (2003)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Frank, M., Zaag, H.: Determination of the blow-up rate for a critical semilinear wave equation. Math. Ann. 331(2), 395–416 (2005)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Olver, F.W.J.: Asymptotics and special functions. Computer Science and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1974)Google Scholar
  36. 36.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST handbook of mathematical functions. U.S. Department of Commerce National Institute of Standards and Technology, Washington, DC: With 1 CD-ROM. Windows, Macintosh and UNIX (2010)Google Scholar
  37. 37.
    Pecher, H.: Nonlinear small data scattering for the wave and Klein-Gordon equation. Math. Z. 185(2), 261–270 (1984)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Perelman, G.: On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré 2(4), 605–673 (2001)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Rauch, J.: I. The \(u^{5}\) Klein-Gordon equation. II. Anomalous singularities for semilinear wave equations. In: Nonlinear partial differential equations and their applications. Collège de France Seminar, vol. I (Paris, 1978/1979). Res. Notes in Math., vol. 53, pp. 335–364. Pitman, Boston (1981)Google Scholar
  40. 40.
    Shatah, J., Struwe, M.: Regularity results for nonlinear wave equations. Ann. Math. (2) 138(3), 503–518 (1993)Google Scholar
  41. 41.
    Shatah, J., Struwe, M.: Well-posedness in the energy space for semilinear wave equations with critical growth. Int. Math. Res. Notices (7):303ff (1994, electronic)Google Scholar
  42. 42.
    Soffer, A.: Soliton dynamics and scattering. In: International Congress of Mathematicians, vol. III, pp. 459–471. Eur. Math. Soc., Zürich (2006)Google Scholar
  43. 43.
    Sogge, C.D.: Lectures on non-linear wave equations, 2nd edn. International Press, Boston (2008)Google Scholar
  44. 44.
    Struwe, M.: Globally regular solutions to the \(u^5\) Klein-Gordon equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15(3), 495–513 (1989, 1988)Google Scholar
  45. 45.
    Szpak, N.: Relaxation to intermediate attractors in nonlinear wave equations. Theor. Math. Phys. 127, 817–826 (2001). doi: 10.1023/A:1010460004007 MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Terence, T.: On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation. Dyn. Partial Diff. Equ. 1(1), 1–48 (2004)MATHGoogle Scholar
  47. 47.
    Terence, T.: Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions. Dyn. Partial Diff. Equ. 3(2), 93–110 (2006)MATHGoogle Scholar
  48. 48.
    Taylor, M.E.: Tools for PDE. Mathematical Surveys and Monographs, vol. 81. Pseudodifferential operators, paradifferential operators, and layer potentials. American Mathematical Society, Providence (2000)Google Scholar
  49. 49.
    Teschl, G.: Mathematical methods in quantum mechanics. Graduate Studies in Mathematics, vol. 99. American Mathematical Society, Providence (2009) (With applications to Schrödinger operators)Google Scholar
  50. 50.
    Weidmann, J.: Lineare Operatoren in Hilberträumen. Teil II. Mathematische Leitfäden. [Mathematical Textbooks]. B. G. Teubner, Stuttgart (2003). Anwendungen. [Applications]Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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