Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A Liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up


Authors: Nejla Nouaili and Hatem Zaag
Journal: Trans. Amer. Math. Soc. 362 (2010), 3391-3434
MSC (2000): Primary 35B05, 35K05, 35K55, 74H35
DOI: https://doi.org/10.1090/S0002-9947-10-04902-0
Published electronically: February 17, 2010
MathSciNet review: 2601595
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a Liouville theorem for a vector valued semilinear heat equation with no gradient structure. Classical tools such as the maximum principle or energy techniques break down and have to be replaced by a new approach. We then derive from this theorem uniform estimates for blow-up solutions of that equation.


References [Enhancements On Off] (What's this?)

  • [AHV97] D. Andreucci, M. A. Herrero, and J. J. L. Velázquez.
    Liouville theorems and blow up behaviour in semilinear reaction diffusion systems.
    Ann. Inst. H. Poincaré Anal. Non Linéaire, 14(1):1-53, 1997. MR 1437188 (98e:35088)
  • [Bal77] J. M. Ball.
    Remarks on blow-up and nonexistence theorems for nonlinear evolution equations.
    Quart. J. Math. Oxford Ser. (2), 28(112):473-486, 1977. MR 0473484 (57:13150)
  • [FK92] S. Filippas and R.V. Kohn.
    Refined asymptotics for the blowup of $ u\sb t-\Delta u=u^ p$.
    Comm. Pure Appl. Math., 45(7):821-869, 1992. MR 1164066 (93g:35066)
  • [FM95] S. Filippas and F. Merle.
    Modulation theory for the blowup of vector-valued nonlinear heat equations.
    J. Differential Equations, 116(1):119-148, 1995. MR 1317705 (95m:35087)
  • [Fuj66] H. Fujita.
    On the blowing up of solutions of the Cauchy problem for $ u\sb{t}=\Delta u+u^{1+\alpha }$.
    J. Fac. Sci. Univ. Tokyo Sect. I, 13:109-124 (1966), 1966. MR 0214914 (35:5761)
  • [GH96] M. Grayson and R. S. Hamilton.
    The formation of singularities in the harmonic map heat flow.
    Comm. Anal. Geom., 4(4):525-546, 1996. MR 1428088 (98g:58034)
  • [GK85] Y. Giga and R.V. Kohn.
    Asymptotically self-similar blow-up of semilinear heat equations.
    Comm. Pure Appl. Math., 38(3):297-319, 1985. MR 784476 (86k:35065)
  • [GMS04] Y. Giga, S. Matsui, and S. Sasayama.
    Blow up rate for semilinear heat equations with subcritical nonlinearity.
    Indiana Univ. Math. J., 53(2):483-514, 2004. MR 2060042 (2005g:35153)
  • [GS81a] B. Gidas and J. Spruck.
    Global and local behavior of positive solutions of nonlinear elliptic equations.
    Comm. Pure Appl. Math., 34(4):525-598, 1981. MR 615628 (83f:35045)
  • [GS81b] B. Gidas and J. Spruck.
    A priori bounds for positive solutions of nonlinear elliptic equations.
    Comm. Partial Differential Equations, 6(8):883-901, 1981. MR 619749 (82h:35033)
  • [Ham95] R.S. Hamilton.
    The formation of singularities in the Ricci flow.
    In Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), pages 7-136. Int. Press, Cambridge, MA, 1995. MR 1375255 (97e:53075)
  • [HV93] M. A. Herrero and J. J. L. Velázquez.
    Blow-up behaviour of one-dimensional semilinear parabolic equations.
    Ann. Inst. H. Poincaré Anal. Non Linéaire, 10(2):131-189, 1993. MR 1220032 (94g:35030)
  • [Lev73] H. A. Levine.
    Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $ Pu\sb{t}=-Au+{ F}(u)$.
    Arch. Rational Mech. Anal., 51:371-386, 1973. MR 0348216 (50:714)
  • [LO96] C. D. Levermore and M. Oliver.
    The complex Ginzburg-Landau equation as a model problem.
    In Dynamical systems and probabilistic methods in partial differential equations (Berkeley, CA, 1994), volume 31 of Lectures in Appl. Math., pages 141-190. Amer. Math. Soc., Providence, RI, 1996. MR 1363028 (96k:35168)
  • [MM00] Y. Martel and F. Merle.
    A Liouville theorem for the critical generalized Korteweg-de Vries equation.
    J. Math. Pures Appl. (9), 79(4):339-425, 2000. MR 1753061 (2001i:37102)
  • [MR04] F. Merle and P. Raphael.
    On universality of blow-up profile for $ L^ 2$ critical nonlinear Schrödinger equation.
    Invent. Math., 156(3):565-672, 2004. MR 2061329 (2006a:35283)
  • [MR05] F. Merle and P. Raphael.
    The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation.
    Ann. of Math. (2), 161(1):157-222, 2005. MR 2150386 (2006k:35277)
  • [MZ98a] F. Merle and H. Zaag.
    Optimal estimates for blowup rate and behavior for nonlinear heat equations.
    Comm. Pure Appl. Math., 51(2):139-196, 1998. MR 1488298 (98k:35107)
  • [MZ98b] F. Merle and H. Zaag.
    Refined uniform estimates at blow-up and applications for nonlinear heat equations.
    Geom. Funct. Anal., 8(6):1043-1085, 1998. MR 1664791 (2000c:35106)
  • [MZ00] F. Merle and H. Zaag.
    A Liouville theorem for vector-valued nonlinear heat equations and applications.
    Math. Ann., 316(1):103-137, 2000. MR 1735081 (2001d:35084)
  • [MZ08a] N. Masmoudi and H. Zaag.
    Blow-up profile for the complex Ginzburg-Landau equation.
    J. Funct. Anal., 255(7):1613-1666, 2008. MR 2442077
  • [MZ08b] F. Merle and H. Zaag.
    Openness of the set of non-characteristic points and regularity of the blow-up curve for the $ 1$ D semilinear wave equation.
    Comm. Math. Phys.,282(1):55-86, 2008. MR 2415473
  • [Nou08] N. Nouaili.
    A simplified proof of a Liouville theorem for nonnegative solution of a subcritical semilinear heat equations.
    J. Dynam. Differential Equations, 2008.
    to appear.
  • [PKK98] S. Popp, O. Stiller, E. Kuznetsov, and L. Kramer.
    The cubic complex Ginzburg-Landau equation for a backward bifurcation.
    Phys. D, 114(1-2):81-107, 1998. MR 1612047 (98k:35181)
  • [PŠ01] P. Plecháč and V. Šverák.
    On self-similar singular solutions of the complex Ginzburg-Landau equation.
    Comm. Pure Appl. Math., 54(10):1215-1242, 2001. MR 1843986 (2002d:35194)
  • [Vel92] J. J. L. Velázquez.
    Higher-dimensional blow up for semilinear parabolic equations.
    Comm. Partial Differential Equations, 17(9-10):1567-1596, 1992. MR 1187622 (93k:35044)
  • [Vel93] J. J. L. Velázquez.
    Classification of singularities for blowing up solutions in higher dimensions.
    Trans. Amer. Math. Soc., 338(1):441-464, 1993. MR 1134760 (93j:35101)
  • [Wei84] F. B. Weissler.
    Single point blow-up for a semilinear initial value problem.
    J. Differential Equations, 55(2):204-224, 1984. MR 764124 (86a:35076)
  • [Zaa98] H. Zaag.
    Blow-up results for vector-valued nonlinear heat equations with no gradient structure.
    Ann. Inst. H. Poincaré Anal. Non Linéaire, 15(5):581-622, 1998. MR 1643389 (99i:35066)
  • [Zaa01] H. Zaag.
    A Liouville theorem and blowup behavior for a vector-valued nonlinear heat equation with no gradient structure.
    Comm. Pure Appl. Math., 54(1):107-133, 2001. MR 1787109 (2001h:35088)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35B05, 35K05, 35K55, 74H35

Retrieve articles in all journals with MSC (2000): 35B05, 35K05, 35K55, 74H35


Additional Information

Nejla Nouaili
Affiliation: Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm 75230, Paris Cedex 05, France
Address at time of publication: Université Paris 13, Institut Galilée, Laboratoire Analyse, Géométrie et Applications, CNRS-UMR 7539, 99 avenue J.B. Clément, 93430 Villetaneuse, France
Email: nouaili@math.univ-paris13.fr

Hatem Zaag
Affiliation: Université Paris 13, Institut Galilée, Laboratoire Analyse, Géométrie et Applications, CNRS-UMR 7539, 99 avenue J.B. Clément, 93430 Villetaneuse, France
Email: zaag@math.univ-paris13.fr

DOI: https://doi.org/10.1090/S0002-9947-10-04902-0
Keywords: Blow-up, Liouville theorem, uniform estimates, heat equation, vector valued
Received by editor(s): October 24, 2007
Published electronically: February 17, 2010
Additional Notes: The authors would like to thank the referee for his valuable suggestions which (we hope) made our paper much clearer and reader friendly.
The second author was supported by a grant from the French Agence Nationale de la Recherche, project ONDENONLIN, reference ANR-06-BLAN-0185.
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society