Skip to main content
Log in

Decay of the Solutions of Parabolic Equations with Double Nonlinearity and the Degenerate Absorption Potential

  • Published:
Ukrainian Mathematical Journal Aims and scope

We study the behavior of solutions for the parabolic equation of nonstationary diffusion with double nonlinearity and a degenerate absorption term:

$$ {\left({\left| u\right|}^{q-1} u\right)}_t-{\displaystyle \sum_{i=1}^N\frac{\partial }{\partial {x}_i}\left({\left|{\nabla}_x u\right|}^{q-1}\frac{\partial u}{\partial {x}_i}\right)+{a}_0(x){\left| u\right|}^{\lambda -1} u=0,} $$

where \( {a}_0(x)\ge {d}_0\; \exp \left(-\frac{\omega \left(\left| x\right|\right)}{{\left| x\right|}^{q+1}}\right) \), d 0 = const > 0, 0 ≤ λ < q, ω(⋅) ϵ C([0, + ∞)), ω(0) = 0, ω(τ) > 0 for τ > 0, and \( {\displaystyle {\int}_{0+}\frac{\omega \left(\tau \right)}{\tau} d\tau <\infty } \). By the local energy method, we show that a Dini-type condition imposed on the function ω(·) guarantees the decay of an arbitrary solution for a finite period of time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. H. W. Alt and S. Luckhaus, “Quasilinear elliptic-parabolic differential equations,” Math. Z., 183, No. 3, 311–341 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  2. F. Bernis, “Existence results for doubly nonlinear higher order parabolic equations on unbounded domain,” Math. Ann., 279, No. 3, 373–394 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  3. L. E. Payne, “Improperly posed problems in partial differential equations,” Reg. Conf. Ser. Appl. Math., No. 22, 76 (1975).

  4. B. F. Knerr, “The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension,” Trans. Amer. Math. Soc., 249, No. 2, 409–424 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  5. B. Straughan, “Instability, nonexistence, and weighted energy methods in fluid dynamics and related theories,” Pitman, London (1982).

    MATH  Google Scholar 

  6. C. Bandle and I. Stakgold, “The formation of the dead core in parabolic reaction-diffusion problems,” Trans. Amer. Math. Soc., 286, No. 1, 275–293 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Friedman and M. A. Herrero, “Extinction properties of semilinear heat equations with strong absorption,” J. Math. Anal. Appl., 124, No. 2, 530–546 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen Xu-Yan, H. Matano, and M. Mimura, “Finite-point extinction and continuity of interfaces in a nonlinear diffusion equation with strong absorption,” J. Reine Angew. Math., 459, No. 1, 1–36 (1995).

    MATH  MathSciNet  Google Scholar 

  9. A. S. Kalashnikov, “On the character of propagation of perturbations in the problems of nonlinear heat conduction with absorption,” Zh. Vychisl. Mat. Mat. Fiz., 14, No. 4, 891–905 (1974).

    MathSciNet  Google Scholar 

  10. P. Benilan and M. G. Crandall, “The continuous dependence on φ of solutions of u t Δφ(u) = 0,Indiana Univ. Math. J., 30, No. 2, 161–177 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  11. G. Diaz and I. Diaz, “Finite extinction time for a class of nonlinear parabolic equations,” Comm. Part. Different. Equat., 4, No. 11, 1213–1231 (1979).

    Article  MATH  Google Scholar 

  12. L. A. Peletier, “The porous media equation,” Appl. Nonlin. Anal. Phys. Sci., Bielefeld (1979); Surv. Ref. Works Math., 6, 229–241 (1981).

  13. F. Bernis, “Finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption,” Proc. Roy. Soc. Edinburgh Sect., 104 A, No. 1-2, 1–19 (1986).

    MathSciNet  Google Scholar 

  14. A. E. Shishkov, “Dead zones and instantaneous compactification of the supports of energy solutions of quasilinear parabolic equations of arbitrary order,” Mat. Sb., 190, No. 12, 129–156 (1999).

    Article  Google Scholar 

  15. A. Shishkov and R. Kersner, “Instantaneous shrinking of the support of energy solutions,” J. Math. Anal. Appl., 198, No. 3, 729–750 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Kersner and F. Nicolosi, “The nonlinear heat equation with absorption: effects of variable coefficients,” J. Math. Anal. Appl., 170, No. 2, 551–566 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  17. A. S. Kalashnikov, “Instantaneous shrinking of the support for solutions to certain parabolic equations and systems,” Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. e Natur. Rend., 8, No. 4, 263–272 (1997).

    MATH  MathSciNet  Google Scholar 

  18. Li Jun-Jie, “Qualitative properties for solutions of semilinear heat equations with strong absorption,” J. Math. Anal. Appl., 281, No. 1, 382–394 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  19. Li Jun-Jie, “Qualitative properties of solutions to semilinear heat equations with singular initial data,” Electron. J. Different. Equat., No. 53, 1–12 (2004).

  20. V. A. Kondratiev and L. Véron, “Asymptotic behavior of solutions of some nonlinear parabolic or elliptic equations,” Asymptot. Anal., 14, 117–156 (1997).

    MATH  MathSciNet  Google Scholar 

  21. Y. Belaud, B. Helffer, and L. Véron, “Long-time vanishing properties of solutions of sublinear parabolic equations and semiclassical limit of Schrödinger operator,” Ann. Inst. H. Poincaré Anal. Nonlin., 18, No. 1, 43–68 (2001).

    Article  MATH  Google Scholar 

  22. Y. Belaud and A. Shishkov, “Long-time extinction of solutions of some semilinear parabolic equations,” J. Different. Equat., 238, 64–86 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  23. J. I. Diaz and L. Veron, “Local vanishing properties of solutions of elliptic and parabolic quasilinear equations,” Trans. Amer. Math. Soc., 290, No. 2, 787–814 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  24. S. N. Antontsev, “On the localization of solutions of nonlinear degenerate elliptic and parabolic equations,” Dokl. Akad. Nauk SSSR, 260, No 6, 1289–1293 (1981).

    MathSciNet  Google Scholar 

  25. E. Gagliardo, “Ulteriori proprieta ’di alcune classi di funzioni in piu’ variabili,” Ric. Mat., 8, 24–51 (1959).

    MATH  MathSciNet  Google Scholar 

  26. L. Nirenberg, “On elliptic partial differential equations,” Ann. Scuola Norm. Super. Pisa, 13, No. 3, 115–162 (1959).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 1, pp. 89–107, January, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stepanova, E.V. Decay of the Solutions of Parabolic Equations with Double Nonlinearity and the Degenerate Absorption Potential. Ukr Math J 66, 99–121 (2014). https://doi.org/10.1007/s11253-014-0915-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-014-0915-x

Keywords

Navigation