Blow Up Profiles for a Quasilinear Reaction–Diffusion Equation with Weighted Reaction with Linear Growth

  • Razvan Gabriel IagarEmail author
  • Ariel Sánchez


We study the blow up profiles associated to the following second order reaction–diffusion equation with non-homogeneous reaction:
$$\begin{aligned} \partial _tu=\partial _{xx}(u^m) + |x|^{\sigma }u, \end{aligned}$$
with \(\sigma >0\). Through this study, we show that the non-homogeneous coefficient \(|x|^{\sigma }\) has a strong influence on the blow up behavior of the solutions. First of all, it follows that finite time blow up occurs for self-similar solutions u, a feature that does not appear in the well known autonomous case \(\sigma =0\). Moreover, we show that there are three different types of blow up self-similar profiles, depending on whether the exponent \(\sigma \) is closer to zero or not. We also find an explicit blow up profile. The results show in particular that global blow up occurs when \(\sigma >0\) is sufficiently small, while for \(\sigma >0\) sufficiently large blow up occurs only at infinity, and we give prototypes of these phenomena in form of self-similar solutions with precise behavior. This work is a part of a larger program of understanding the influence of non-homogeneous weights on the blow up sets and rates.


Reaction–diffusion equations Non-homogeneous reaction Blow up Critical case Self-similar solutions Phase space analysis 

Mathematics Subject Classification

35B33 35B40 35K10 35K67 35Q79 



R.I. is supported by the ERC Starting Grant GEOFLUIDS 633152. A.S. is partially supported by the Spanish Project MTM2017-87596-P.


  1. 1.
    Andreucci, D., Tedeev, A.F.: Universal bounds at the blow-up time for nonlinear parabolic equations. Adv. Differ. Equ. 10(1), 89–120 (2005)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bai, X., Zhou, S., Zheng, S.: Cauchy problem for fast diffusion equation with localized reaction. Nonlinear Anal. 74(7), 2508–2514 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bandle, C., Levine, H.: On the existence and nonexistence of global solutions of reaction–diffusion equations in sectorial domains. Trans. Am. Math. Soc. 316, 595–622 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baras, P., Kersner, R.: Local and global solvability of a class of semilinear parabolic equations. J. Differ. Equ. 68, 238–252 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chow, S.N., Hale, J.K.: Methods of Bifurcation Theory. Springer, New York (1982)CrossRefzbMATHGoogle Scholar
  6. 6.
    de Pablo, A., Sánchez, A.: Self-similar solutions satisfying or not the equation of the interface. J. Math. Anal. Appl. 276(2), 791–814 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ferreira, R., de Pablo, A., Vázquez, J.L.: Classification of blow-up with nonlinear diffusion and localized reaction. J. Differ. Equ. 231(1), 195–211 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Galaktionov, V.A., Vázquez, J.L.: Continuation of blowup solutions of nonlinear heat equations in several space dimensions. Commun. Pure Appl. Math. 50(1), 1–67 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Giga, Y., Umeda, N.: Blow-up directions at space infinity for solutions of semilinear heat equations. Bol. Soc. Paran. Mat. 23, 9–28 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Giga, Y., Umeda, N.: On blow-up at space infinity for semilinear heat equations. J. Math. Anal. Appl. 316, 538–555 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gilding, B.H., Peletier, L.A.: On a class of similarity solutions of the porous media equation. J. Math. Anal. Appl. 55, 351–364 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guo, J.-S., Lin, C.-S., Shimojo, M.: Blow-up behavior for a parabolic equation with spatially dependent coefficient. Dyn. Syst. Appl. 19(3–4), 415–433 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Guo, J.-S., Shimojo, M.: Blowing up at zero points of potential for an initial boundary value problem. Commun. Pure Appl. Anal. 10(1), 161–177 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guo, J.-S., Lin, C.-S., Shimojo, M.: Blow-up for a reaction–diffusion equation with variable coefficient. Appl. Math. Lett. 26(1), 150–153 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Iagar, R.G., Laurençot, Ph: Existence and uniqueness of very singular solutions for a fast diffusion equation with gradient absorption. J. Lond. Math. Soc. 87, 509–529 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Iagar, R.G., Laurençot, Ph: Self-similar extinctionfor a diffusive Hamilton–Jacobi equation with critical absorption. Calc. Var. PDE 56(3), 1–38 (2017). (Art. 77)CrossRefzbMATHGoogle Scholar
  17. 17.
    Iagar, R.G., Sánchez, A.: Blow up profiles for a quasilinear reaction–diffusion equation with weighted reaction. Preprint arXiv:1811.10330 (2018)
  18. 18.
    Igarashi, T., Umeda, N.: Existence and nonexistence of global solutions in time for a reaction–diffusion system with inhomogeneous terms. Funkc. Ekvac. 51(1), 17–37 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kang, X., Wang, W., Zhou, X.: Classification of solutions of porous medium equation with localized reaction in higher space dimensions. Differ. Integral Equ. 24(9–10), 909–922 (2011)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lacey, A.A.: The form of blow-up for nonlinear parabolic equations. Proc. R. Soc. Edinb. Sect. A 98(1–2), 183–202 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liang, Z.: On the critical exponents for porous medium equation with a localized reaction in high dimensions. Commun. Pure Appl. Anal. 11(2), 649–658 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lyagina, L.S.: The integral curves of the equation \(y^{\prime }=\frac{ax^2+bxy+cy^2}{dx^2+exy+fy^2}\). Uspekhi Mat. Nauk 6(2), 171–183 (1951). (Russian)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Perko, L.: Differential Equations and Dynamical Systems. Texts in Applied Mathematics, vol. 7, 3rd edn. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  24. 24.
    Pinsky, R.G.: Existence and nonexistence of global solutions for \(u_t=\Delta u+a(x)u^p\) in \({\mathbb{R}}^d\). J. Differ. Equ. 133(1), 152–177 (1997)CrossRefGoogle Scholar
  25. 25.
    Pinsky, R.G.: The behavior of the life span for solutions to \(u_t=\Delta u+a(x)u^p\) in \({\mathbb{R}}^d\). J. Differ. Equ. 147(1), 30–57 (1998)CrossRefzbMATHGoogle Scholar
  26. 26.
    Quittner, P., Souplet, P.: Superlinear Parabolic Problems. Blow-Up, Global Existence And Steady States, Birkhauser Advanced Texts. Birkhauser Verlag, Basel (2007)zbMATHGoogle Scholar
  27. 27.
    Samarskii, A.A., Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P.: Blow-Up in Quasilinear Parabolic Problems. de Gruyter Expositions in Mathematics, vol. 19. W. de Gruyter, Berlin (1995)CrossRefGoogle Scholar
  28. 28.
    Suzuki, R.: Existence and nonexistence of global solutions of quasilinear parabolic equations. J. Math. Soc. Jpn. 54(4), 747–792 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Instituto de Ciencias Matemáticas (ICMAT)MadridSpain
  2. 2.Institute of Mathematics of the Romanian AcademyBucharestRomania
  3. 3.Departamento de Matemática Aplicada, Ciencia e Ingenieria de los Materiales y Tecnologia ElectrónicaUniversidad Rey Juan CarlosMóstolesSpain

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