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

Abstract

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.

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References

  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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chow, S.N., Hale, J.K.: Methods of Bifurcation Theory. Springer, New York (1982)

    Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Perko, L.: Differential Equations and Dynamical Systems. Texts in Applied Mathematics, vol. 7, 3rd edn. Springer, New York (2001)

    Google 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)

    Article  Google 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)

    Article  Google Scholar 

  26. 26.

    Quittner, P., Souplet, P.: Superlinear Parabolic Problems. Blow-Up, Global Existence And Steady States, Birkhauser Advanced Texts. Birkhauser Verlag, Basel (2007)

    Google 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)

    Google Scholar 

  28. 28.

    Suzuki, R.: Existence and nonexistence of global solutions of quasilinear parabolic equations. J. Math. Soc. Jpn. 54(4), 747–792 (2002)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

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

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Correspondence to Razvan Gabriel Iagar.

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Iagar, R.G., Sánchez, A. Blow Up Profiles for a Quasilinear Reaction–Diffusion Equation with Weighted Reaction with Linear Growth. J Dyn Diff Equat 31, 2061–2094 (2019). https://doi.org/10.1007/s10884-018-09727-w

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Keywords

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

Mathematics Subject Classification

  • 35B33
  • 35B40
  • 35K10
  • 35K67
  • 35Q79