Mathematische Annalen

, Volume 364, Issue 1–2, pp 269–292 | Cite as

Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure

Article

Abstract

We provide a simple method for obtaining new Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. To illustrate the method we prove Liouville theorems (guaranteeing nonexistence of positive classical solutions) for the following model problems: the scalar nonlinear heat equation
$$\begin{aligned} u_t-\Delta u=u^p \qquad \hbox {in }\ {\mathbb R}^n\times {\mathbb R}, \end{aligned}$$
its vector-valued generalization with a \(p\)-homogeneous nonlinearity and the linear heat equation in \({\mathbb R}^n_+\times {\mathbb R}\) complemented by nonlinear boundary conditions of the form \(\partial u/\partial \nu =u^q\). Here \(\nu \) denotes the outer unit normal on the boundary of the halfspace \({\mathbb R}^n_+\) and the exponents \(p,q>1\) satisfy \(p<n/(n-2)\) and \(q<(n-1)/(n-2)\) if \(n>2\) (or \(p<(n+2)/(n-2)\) and \(q<n/(n-2)\) if \(n>2\) and some symmetry of the solutions is assumed). As a typical application of our nonexistence results we provide optimal universal estimates for positive solutions of related problems in bounded and unbounded domains.

Mathematics Subject Classification

35K55 35K15 35K20 35K60 35K40 35B45 35B40 

References

  1. 1.
    Bartsch, T., Poláčik, P., Quittner, P.: Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations. J. Eur. Math. Soc. 13, 219–247 (2011)CrossRefMATHGoogle Scholar
  2. 2.
    Bidaut-Véron, M.-F.: Initial blow-up for the solutions of a semilinear parabolic equation with source term. Equations aux dérivées partielles et applications. articles dédiés à Jacques-Louis Lions, pp. 189–198. Gauthier-Villars, Paris (1998)Google Scholar
  3. 3.
    Chlebík, M., Fila, M.: On the blow-up rate for the heat equation with a nonlinear boundary condition. Math. Methods Appl. Sci. 23, 1323–1330 (2000)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Dancer, N.: Some notes on the method of moving planes. Bull. Austral. Math. Soc. 46, 425–434 (1992)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Deng, K., Fila, M., Levine, H.A.: On critical exponents for a system of heat equations coupled in the boundary conditions. Acta Math. Univ. Comenianae 63, 169–192 (1994)MathSciNetMATHGoogle Scholar
  6. 6.
    Eidelman, S.D.: Estimates of solutions of parabolic systems and some of their applications. Math. Sbornik 33, 359–382 (1953). (in Russian)MathSciNetGoogle Scholar
  7. 7.
    Eidelman, S.D.: Liouville-type theorems for parabolic and elliptic systems. Doklady AN SSSR 99, 681–684 (1954). (in Russian)MathSciNetGoogle Scholar
  8. 8.
    Fila, M., Souplet, Ph, Weissler, F.B.: Linear and nonlinear heat equations in \(L^p_\delta \) spaces and universal bounds for global solutions. Math. Ann. 320, 87–113 (2001)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Fila, M., Yanagida, E.: Homoclinic and heteroclinic orbits for a semilinear parabolic equation. Tohoku Math. J. 63, 561–579 (2011)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Fujita, H.: On the blowing up of solutions of the Cauchy problem for \(u_t=\Delta u+u^{1+\alpha }\). J. Fac. Sci. Univ. Tokyo Sec. IA Math. 13, 109–124 (1966)MATHGoogle Scholar
  11. 11.
    Galaktionov, V.A., Levine, H.A.: On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary. Israel J. Math. 94, 125–146 (1996)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34, 525–598 (1981)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Giga, Y., Kohn, R.: Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36, 1–40 (1987)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Harada, J.: Positive solutions to the Laplace equation with nonlinear boundary conditions on the half space. Calc. Var. 50, 399–435 (2014)CrossRefMATHGoogle Scholar
  15. 15.
    Harada, J.: Non self-similar blow-up solutions to the heat equation with nonlinear boundary conditions. Nonlinear Anal. TMA 102, 36–83 (2014)CrossRefMATHGoogle Scholar
  16. 16.
    Hayakawa, K.: On nonexistence of global solutions of some semilinear parabolic differential equations. Proc. Japan Acad. 49, 503–505 (1973)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Hu, B.: Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition. Differ. Integral Equations 7, 301–313 (1994)MATHGoogle Scholar
  18. 18.
    Hu, B.: Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition. Differ. Integral Equations 9, 891–901 (1996)MATHGoogle Scholar
  19. 19.
    Hu, B., Yin, H.-M.: The profile near blowup time for solution of the heat equation with a nonlinear boundary condition. Trans. Amer. Math. Soc. 346, 117–135 (1994)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Ishige, K., Sato, R.: Heat equation with a nonlinear boundary condition and uniformly local \(L^r\) spaces. Preprint arXiv:1404.6856
  21. 21.
    Kobayashi, K., Sirao, T., Tanaka, H.: On the blowing up problem for semilinear heat equations. J. Math. Soc. Japan 29, 407–424 (1977)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Merle, F., Zaag, H.: Optimal estimates for blowup rate and behavior for nonlinear heat equations. Comm. Pure Appl. Math. 51, 139–196 (1998)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Nicolesco, M.: Sur l’equation de la chaleur. Comm. Math. Helvetici 10, 3–17 (1937)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Phan, Q.H.: Optimal Liouville-type theorems for a parabolic system. Discrete Contin. Dynam. Syst. 35, 399–409 (2015)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Poláčik, P., Quittner, P.: A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation. Nonlinear Anal. 64, 1679–1689 (2006)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Poláčik, P., Quittner, P., Souplet, Ph: Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems. Duke Math. J. 139, 555–579 (2007)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Poláčik, P., Quittner, P., Souplet, Ph: Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: parabolic equations. Indiana Univ. Math. J. 56, 879–908 (2007)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Quittner, P., Souplet, Ph.: Superlinear parabolic problems. Blow-up, global existence and steady states. Birkhäuser Advanced Texts, Birkhäuser, Basel (2007)Google Scholar
  29. 29.
    Quittner, P., Souplet, Ph.: Parabolic Liouville-type theorems via their elliptic counterparts. Discrete Contin. Dynam. Systems, Supplement 2011. (Proceedings of the 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, Dresden 2010), pp. 1206–1213 (2011)Google Scholar
  30. 30.
    Quittner, P., Souplet, Ph: Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete Contin. Dynam. Systems S 5, 671–681 (2012)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Quittner, P., Souplet, Ph: Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications. Comm. Math. Phys. 311, 1–19 (2012)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics and StatisticsComenius UniversityBratislavaSlovakia

Personalised recommendations