Advertisement

Solving a nonlinear variation of the heat equation: self-similar solutions of the second kind and other results

  • Rodrigo MenesesEmail author
  • Oscar Orellena
Article
  • 15 Downloads

Abstract

This paper studies regular self-similar solutions of the following diffusion equation
$$\begin{aligned} u_{t}+\gamma \vert u_{t} \vert =\Delta u\quad \text {in}\ \mathbb {R}^{N}\times ]0,\infty [, \end{aligned}$$
where \(-1<\gamma <1\). The analysis is focused on radial symmetric solutions \(u(x,t)=t^{-\alpha /2}f(\eta )\) with \(\alpha >0\) and \(\eta =\Vert x\Vert /\sqrt{t}\). Closed representation is obtained in terms of confluent hypergeometric functions. Employing specific properties of these special functions, oscillatory and symptotic aspects of f are obtained. It is demonstrated that such features are governed by increasing and unbounded sequences of exponents \(\alpha _{0}<\alpha _{1}<\cdots \), as in other diffusion equations. These exponents are determined by solving a system of transcendental equations related to specific roots of Kummer and Tricomi functions. As these cannot be determined using dimensional analysis, it is concluded that they are anomalous. For each exponent \(\alpha _{k}\), linear approximation when \(\gamma \) is close to zero is also presented. Finally, relationships with previous results as well as an extension to other fully nonlinear parabolic equations are discussed.

Keywords

Fully nonlinear parabolic equations Second kind self-similar solutions Anomalous exponents Confluent hypergeometric functions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The first author would like to thank J. F. Jabir and P. Quintana G. for their help with the redaction of parts of this article. The work of the second author has been supported by Fondecyt (Chile) Grant 1181414.

References

  1. 1.
    Abramowitz, M. & Stegun, I.: Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Courier Dover Publications (1972)Google Scholar
  2. 2.
    Aronson, D. & Vázquez, J.: Calculation of anomalous exponents in nonlinear diffusion. Physical Review Letters. 72(3), 348–351 (1994)CrossRefGoogle Scholar
  3. 3.
    Barenblatt, G.: Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics, volume 14. Cambridge University Press (1996)Google Scholar
  4. 4.
    Barenblatt, G. & Krylov, A. Concerning the elastico-plastic regime of filtration. Izv. Akad. Nauk SSSR, OTN. 2, 5–13 (1955)Google Scholar
  5. 5.
    Barenblatt, G. & Sivashinskii, G. Self-similar solutions of the second kind in nonlinear filtration. Journal of Applied Mathematics and Mechanics. 33(5), 836–845 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Caffarelli, L. & Stefanelli, U.: A counterexample to \(\cal{C}^{2,1}\) regularity for parabolic fully nonlinear equations. Comm. in Partial Differential Equations. 33(7), 1216–1234 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    van Duijn, C., Gomes, S. & Zhang, H.: On a class of similarity solutions of the equation \(u_{t}=(\vert u\vert ^{m-1} u_{x})_{x}\) with \(m>-1\). IMA J. Appl. Math. 41, 147–163 (1988)Google Scholar
  8. 8.
    Eggers, J. & Fontelos, M.: The role of self-similarity in singularities of partial differential equations. Nonlinearity, 22(1):R1 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fleming, W. & Soner. H.: Controlled Markov processes and viscosity solutions, 25, Springer (2006)Google Scholar
  10. 10.
    Galaktionov, V. & Svirshchevskii, S.: Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics. Chapman & Hall/CRC Applied mathematics and nonlinear science series (2007)Google Scholar
  11. 11.
    Giga, M., Giga, Y. & Saal, J.: Nonlinear partial differential equations: Asymptotic behavior of solutions and self-similar solutions, volume 79. Springer (2010)Google Scholar
  12. 12.
    Gradshteyn, I. S. & Ryzhik, I. M.: Table of integrals, series and products, seventh edition. Elsevier Inc (2007)Google Scholar
  13. 13.
    Huang, Y. & Vazquez, J. L.: Large-time geometrical properties of solutions of the Barenblatt equation of elasto-plastic filltration. Journal of Differential Equations, 252(7), 4229–4242 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hulshof, J.: Similarity solutions of the porous medium with sign changes. J. Math. Anal. Appl. vol. 157, 75–111 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hulshof, J., King, J. & Bowen, M.: Intermediate asymptotics of the porous medium equation with sign changes. Adv. Diff.Eq., 6, 1115–1152 ( 2001)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Iagar, R. Sánchez, A. & Vázquez, J. L.: Radial equivalence for the two basic nonlinear diffusion equations. Jour. Math. Pures Appl. 89, 1–24 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Imbert, C. & Silvestre, L. (2013) An introduction to fully nonlinear parabolic equation, in: An introduction to the Kahler-Ricci flow, vol. 2086 in Lectures-Notes in Math. Springer: 7–88.Google Scholar
  18. 18.
    Lamb G. L.: Elements of soliton theory. John Wiley, New York (1980)zbMATHGoogle Scholar
  19. 19.
    Kamenomostskaia, S. L.: On a problem in the theory of filtration. Dokl. Akad. Nauk SSSR, 116, 18–20 (1957)MathSciNetGoogle Scholar
  20. 20.
    Kamin, S., Peletier, L. A. & Vazquez, J. L.: On the Barenblatt equation of elastoplastic filtration. Indiana Univ. Math. J, 40(4), 1333–1362 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Olver, F. J.: Introduction to asymptotics and special functions. Academic Press (1974)Google Scholar
  22. 22.
    Polyanin, A. & Zaitzev, A.: Handbook of nonlinear partial differential equations, second edition. Chapman & Hall/CRC Press, Boca Raton-London-New York (2012)Google Scholar
  23. 23.
    Schlichting, H, Gersten, K.: Boundary layer theory. Springer-Verlag Berlin Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  24. 24.
    Vázquez, J. L.: Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type. Oxford University press Inc. New York (2006)CrossRefzbMATHGoogle Scholar
  25. 25.
    Volpert, A. I., Volpert, V. A. & Volpert, V. A.: Traveling wave solutions of parabolic equations. American mathematical society, Providence, R.I (1994)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Escuela de Ingeniería Civil, Facultad de IngenieríaUniversidad de ValparaísoValparaísoChile
  2. 2.Departamento de MatemáticasUniversidad Técnica Federico Santa MaríaValparaísoChile

Personalised recommendations