Abstract
This paper studies regular self-similar solutions of the following diffusion equation
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.
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04 April 2019
In the original publication, the second author’s surname was incorrect. The correct name should be Oscar Orellana.
References
Abramowitz, M. & Stegun, I.: Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Courier Dover Publications (1972)
Aronson, D. & Vázquez, J.: Calculation of anomalous exponents in nonlinear diffusion. Physical Review Letters. 72(3), 348–351 (1994)
Barenblatt, G.: Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics, volume 14. Cambridge University Press (1996)
Barenblatt, G. & Krylov, A. Concerning the elastico-plastic regime of filtration. Izv. Akad. Nauk SSSR, OTN. 2, 5–13 (1955)
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)
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)
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)
Eggers, J. & Fontelos, M.: The role of self-similarity in singularities of partial differential equations. Nonlinearity, 22(1):R1 (2009)
Fleming, W. & Soner. H.: Controlled Markov processes and viscosity solutions, 25, Springer (2006)
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)
Giga, M., Giga, Y. & Saal, J.: Nonlinear partial differential equations: Asymptotic behavior of solutions and self-similar solutions, volume 79. Springer (2010)
Gradshteyn, I. S. & Ryzhik, I. M.: Table of integrals, series and products, seventh edition. Elsevier Inc (2007)
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)
Hulshof, J.: Similarity solutions of the porous medium with sign changes. J. Math. Anal. Appl. vol. 157, 75–111 (1991)
Hulshof, J., King, J. & Bowen, M.: Intermediate asymptotics of the porous medium equation with sign changes. Adv. Diff.Eq., 6, 1115–1152 ( 2001)
Iagar, R. Sánchez, A. & Vázquez, J. L.: Radial equivalence for the two basic nonlinear diffusion equations. Jour. Math. Pures Appl. 89, 1–24 (2008)
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.
Lamb G. L.: Elements of soliton theory. John Wiley, New York (1980)
Kamenomostskaia, S. L.: On a problem in the theory of filtration. Dokl. Akad. Nauk SSSR, 116, 18–20 (1957)
Kamin, S., Peletier, L. A. & Vazquez, J. L.: On the Barenblatt equation of elastoplastic filtration. Indiana Univ. Math. J, 40(4), 1333–1362 (1991)
Olver, F. J.: Introduction to asymptotics and special functions. Academic Press (1974)
Polyanin, A. & Zaitzev, A.: Handbook of nonlinear partial differential equations, second edition. Chapman & Hall/CRC Press, Boca Raton-London-New York (2012)
Schlichting, H, Gersten, K.: Boundary layer theory. Springer-Verlag Berlin Heidelberg (2000)
Vázquez, J. L.: Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type. Oxford University press Inc. New York (2006)
Volpert, A. I., Volpert, V. A. & Volpert, V. A.: Traveling wave solutions of parabolic equations. American mathematical society, Providence, R.I (1994)
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.
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The original version of this article was revised: The second author’s surname was incorrect and it has been corrected now.
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Meneses, R., Orellana, O. Solving a nonlinear variation of the heat equation: self-similar solutions of the second kind and other results. J. Evol. Equ. 19, 915–929 (2019). https://doi.org/10.1007/s00028-019-00480-1
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DOI: https://doi.org/10.1007/s00028-019-00480-1