Around a singular solution of a nonlocal nonlinear heat equation

  • Piotr BilerEmail author
  • Dominika Pilarczyk
Open Access


We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared in the pointwise sense to the singular solution or in the norm of a critical Morrey space). Then, asymptotics of subcritical solutions is determined. These results are compared with conditions on the initial data leading to a finite time blowup.


Fractional Laplacian Nonlinear heat equation Singular solution Global-in-time solutions Singular potential Asymptotic behavior Stability 

Mathematics Subject Classification

35K55 35B05 35B40 60J60 



  1. 1.
    Alfaro, M.: Fujita blow up phenomena and hair trigger effect: the role of dispersal tails. Ann. Inst. Henri Poincaré, Analyse non Linéaire 34, 1309–1327 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andreucci, D., DiBenedetto, E.: On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 18, 363–441 (1991)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Baras, P., Pierre, M.: Critère d’existence de solutions positives pour des équations semi-linéaires non monotones. Ann. Inst. Henri Poincaré, Anal. non Linéaire 2, 185–212 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barrios, B., Peral, I., Soria, F., Valdinoci, E.: A Widder’s type theorem for the heat equation with nonlocal diffusion. Arch. Ration. Mech. Anal. 213, 629–650 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Biler, P.: The Cauchy problem and self-similar solutions for a nonlinear parabolic equation. Studia Math. 114, 181–205 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Biler, P.: Blowup versus global in time existence of solutions for nonlinear heat equations, 1–13. Topol. Methods Nonlinear Anal. 52, 147–160 (2018)MathSciNetGoogle Scholar
  7. 7.
    Biler, P.: Blowup of solutions for nonlinear nonlocal heat equations. pp. 1–12, Monatsh. Math., to appear; arXiv:1807.03569
  8. 8.
    Biler, P.: Singularities of Solutions to Chemotaxis Systems, book in preparation, De Gruyter, Series in Mathematics and Life SciencesGoogle Scholar
  9. 9.
    Biler, P., Funaki, T., Woyczyński, W.: Fractal Burgers equations. J. Differ. Equ. 148, 9–46 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Biler, P., Imbert, C., Karch, G.: Nonlocal porous medium equation: Barenblatt profiles and other weak solutions. Arch. Ration. Mech. Anal. 215, 497–529 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Biler, P., Karch, G., Woyczyński, W.A.: Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws. Ann. Inst. Henri Poincaré, Anal. non Linéaire 18, 613–637 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Biler, P., Karch, G., Woyczyński, W.A.: Asymptotics for conservation laws involving Lévy diffusion generators. Studia Math. 148, 171–192 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Biler, P., Karch, G., Zienkiewicz, J.: Large global-in-time solutions to a nonlocal model of chemotaxis. Adv. Math. 330, 834–875 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Biler, P., Karch, G., Pilarczyk, D.: Global radial solutions in classical Keller–Segel chemotaxis model. pp. 1–20, submitted; arXiv:1807.02628
  15. 15.
    Biler, P., Zienkiewicz, J.: Blowing up radial solutions in the minimal Keller–Segel chemotaxis model. J. Evol. Equ. 1–20 (2018).
  16. 16.
    Birkner, M., López-Mimbela, J.A., Wakolbinger, A.: Comparison results and steady states for the Fujita equation with fractional Laplacian. Ann. Inst. Henri Poincaré, Anal. non Linéaire 22, 83–97 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bogdan, K., Grzywny, T., Jakubowski, T., Pilarczyk, D.: Fractional Laplacian with Hardy potential. pp. 1–28; arXiv:1710.08378. Commun. Partial Differ. Equ. (2019).
  18. 18.
    Bonforte, M., Sire, Y., Vázquez, J.L.: Optimal existence and uniqueness theory for the fractional heat equation. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 153, 142–168 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Burczak, J., Granero-Belinchón, R.: Global solutions for a supercritical drift–diffusion equation. Adv. Math. 295, 334–367 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Celik, C., Zhou, Z.: No local \(L^1\) solution for a nonlinear heat equation. Commun. Partial Differ. Equ. 28, 1807–1831 (2003)CrossRefGoogle Scholar
  21. 21.
    Constantin, P., Vicol, V.: Nonlinear maximum principles for dissipative linear nonlocal operators and applications. Geom. Funct. Anal. 22, 1289–1321 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249, 511–528 (2004)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fila, M., King, J.R., Winkler, M., Yanagida, E.: Optimal lower bound of the grow-up rate for a supercritical parabolic equation. J. Differ. Equ. 228, 339–356 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Fila, M., Winkler, M.: Rate of convergence to a singular steady state of a supercritical parabolic equation. J. Evol. Equ. 8, 673–692 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Fila, M., Winkler, M., Yanagida, E.: Grow-up rate of solutions for a supercritical semilinear diffusion equation. J. Differ. Equ. 205, 365–389 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Fila, M., Winkler, M., Yanagida, E.: Slow convergence to zero for a parabolic equation with a supercritical nonlinearity. Math. Ann. 340, 477–496 (2008)MathSciNetCrossRefGoogle Scholar
  27. 27.
    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 Sect. I(13), 109–124 (1966)Google Scholar
  28. 28.
    Galaktionov, V.A., Vázquez, J.L.: Continuation of blowup solutions of nonlinear heat equations in several space dimensions. Commun. Pure Appl. Math. 50, 1–67 (1997)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Giga, Y., Miyakawa, T.: Navier–Stokes flow in \({\mathbb{R}}^d\) with measures as initial vorticity and Morrey spaces. Commun. Partial Differ. Equ. 14, 577–618 (1989)CrossRefGoogle Scholar
  30. 30.
    Granero-Belinchón, R., Orive-Illera, R.: An aggregation equation with a nonlocal flux. Nonlinear Anal., Theory Methods Appl., Ser. A 108, 260–274 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Gui, C., Ni, W.-M., Wang, X.: On the stability and instability of positive steady states of a semilinear heat equation in \(\mathbb{R}^n\). Commun. Pure Appl. Math. 45, 1153–1181 (1992)CrossRefGoogle Scholar
  32. 32.
    Lemarié-Rieusset, P.-G.: Recent Developments in the Navier–Stokes Problem. Chapman & Hall/CRC Research Notes in Mathematics 431, Boca Raton (2002)Google Scholar
  33. 33.
    Lemarié-Rieusset, P.-G.: Small data in an optimal Banach space for the parabolic-parabolic and parabolic–elliptic Keller–Segel equations in the whole space. Adv. Differ. Equ. 18, 1189–1208 (2013)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Lemarié-Rieusset, P.-G.: Sobolev multipliers, maximal functions and parabolic equations with a quadratic nonlinearity. J. Funct. Anal. 274, 659–694 (2018)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Mizoguchi, N.: On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity. Math. Z. 239, 215–229 (2002)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Mizoguchi, N.: Boundedness of global solutions for a supercritical semilinear heat equation and its application. Indiana Univ. Math. J. 54, 1047–1059 (2005)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Pilarczyk, D.: Asymptotic stability of singular solution to nonlinear heat equation. Disc. Cont. Dyn. Syst. 25, 991–1001 (2009)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Pilarczyk, D.: Self-similar asymptotics of solutions to heat equation with inverse square potential. J. Evol. Equ. 13, 69–87 (2013)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Poláčik, P., Yanagida, E.: On bounded and unbounded global solutions of a supercritical semilinear heat equation. Math. Ann. 327, 745–771 (2003)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Quittner, P., Souplet, P.: Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States. Birkhäuser Advanced Texts, Basel (2007)Google Scholar
  41. 41.
    Souplet, Ph: Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in \(\mathbb{R}^n\). J. Funct. Anal. 272, 2005–2037 (2017)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Souplet, Ph, Weissler, F.B.: Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state. Ann. Inst. Henri Poincaré, Anal. non Linéaire 20, 213–235 (2003)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Sugitani, S.: On nonexistence of global solutions for some nonlinear integral equations. Osaka J. Math. 12, 45–51 (1975)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Taylor, M.E.: Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations. Commun. Partial Differ. Equ. 17, 1407–1456 (1992)MathSciNetCrossRefGoogle Scholar

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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland
  2. 2.Wydział MatematykiPolitechnika WrocławskaWrocławPoland

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