Journal of Mathematical Sciences

, Volume 234, Issue 1, pp 106–116 | Cite as

Limiting profile of solutions of quasilinear parabolic equations with flat peaking

  • Yevgeniia A. YevgenievaEmail author


The paper deals with energy (weak) solutions u (t; x) of the class of equations with the model representative
$$ \left(\left|u\right|{p}^{-1}u\right)t-\Delta p(u)=0,\kern0.5em \left(t,x\right)\in \left(0,T\right)\times \varOmega, \varOmega \in {\mathrm{\mathbb{R}}}^n,n\ge 1,p>0, $$
and with the following blow-up condition for the energy:
$$ \varepsilon (t):= {\int}_{\Omega}{\left|u\left(t,x\right)\right|}^{p+1} dx+{\int}_0^t{\int}_{\Omega}{\left|{\nabla}_xu\left(\tau, x\right)\right|}^{p+1} dx d\tau \to \infty \mathrm{as}\;t\to T, $$
where Ω is a smooth bounded domain. In the case of flat peaking, namely, under the condition
$$ {\displaystyle \begin{array}{cc}\varepsilon (t)\le F\upalpha (t){\upomega}_0{\left(T-t\right)}^{-\upalpha}& \forall t<T,\end{array}}{\upomega}_0>0,\upalpha >\frac{1}{p+1}, $$

a sharp estimate of the profile of a solution has been obtained in a neighborhood of the blow-up time t = T.


Quasilinear parabolic equations blow-up regime energy solution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. A. Galaktionov and A. A. Samarskii, “Methods of construction of approximate self-similar solutions of nonlinear heat equations,” Mat. Sborn., 118(160), No. 3, 291–322 (1982).Google Scholar
  2. 2.
    B. H. Gilding and M. A. Herrero, “Localization and blow-up of thermal waves in nonlinear heat conduction with peaking,” Math. Ann., 282, No. 2, 223–242 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    C. Cortazar and M. Elgueta, “Localization and boundedness of the solutions of the Neumann problem for a filtration equation,” Nonlinear Anal., 13, No. 1, 33–41 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Regimes with Peaking in Problems for Quasilinear Parabolic Equations [in Russian], Nauka, Moscow, 1987.Google Scholar
  5. 5.
    B. H. Gilding and J. Goncerzewicz, “Localization of solutions of exterior domain problems for the porous media equation with radial symmetry,” SIAM J. Math. Anal., 31, No. 4, 862–893 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. E. Shishkov and A. G. Shchelkov, “Boundary regimes with peaking for general quasilinear parabolic equations in multidimensional domains,” Mat. Sborn., 190, Nos. 3-4, 447–479 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    V. A. Galaktionov and A. E. Shishkov, “Saint-Venant’s principle in blow-up for higher order quasilinear parabolic equations,” Proc. Roy. Soc. Edinburgh. Sect. A, 133, No. 5, 1075–1119 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    V. A. Galaktionov and A. E. Shishkov, “Structure of boundary blow-up for higher-order quasilinear parabolic equations,” Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 460, No. 2051, 3299–3325 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    V. A. Galaktionov and A. E. Shishkov, “Self-similar boundary blow-up for higher-order quasilinear parabolic equations,” Proc. Roy. Soc. Edinburgh, 135A, 1195–1227 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    V. A. Galaktionov and A. E. Shishkov, “Higher-order quasilinear parabolic equations with singular initisl data,” Comm. Contemp. Math., 8, No. 3, 331–354 (2006).CrossRefzbMATHGoogle Scholar
  11. 11.
    A. A. Kovalevsky, I. I. Skrypnik, and A.E. Shishkov, Singular Solutions in Nonlinear Elliptic and Parabolic Equations, De Gruyter, Basel, 2016.CrossRefzbMATHGoogle Scholar
  12. 12.
    A. E. Shishkov and Ye. A. Evgenieva, “Localized peaking regimes for quasilinear parabolic equations,” (2018).
  13. 13.
    G. Stampacchia, Èquations Elliptiques du Second Ordre á Coefficients Discontinus, Univ. Montreal, Montreal, 1966.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and Mechanics of the NAS of UkraineSlavyanskUkraine

Personalised recommendations