Homogeneous diffusion in \({\mathbb{R}}\) with power-like nonlinear diffusivity. (English) Zbl 0683.76073

Summary: We study the nonnegativ solutions of the initial-value problem \[ u_ t = (u^ r | u_ x|^{p-1} u_ x)_ x, \qquad u(x,0)\in L^ 1(\mathbb{R}),\tag{1} \] where \(p>0\), \(r+p>0\). The local velocity of propagation of the solutions is identified as \(V=-v_ x| v_ x|^{p-1}\), where \(v=cu^{\alpha}\) (with \(\alpha = (r+p-1)/p\) and \(c=(r+p)/(r+p-1)\)) is the nonlinear potential. Our main result is the a priori estimate \((v_ x| v_ x|^{p-1})_ x \geq -1/(2p+r)t\) which we use to establish:
i) existence and uniqueness of a solution of (1),
ii) regularity of the free boundaries that appear when \(r+p>1\), and
iii) asymptotic behavior of solutions and free boundaries for initial data with compact support.


76R50 Diffusion
35K57 Reaction-diffusion equations
Full Text: DOI


[1] Ahmed, N., & Sunada, D. K. Nonlinear flows in porous media. J. Hydraulics Div. Proc. Amer. Soc. Civil Eng. 95 (1969), 1847-1857.
[2] Antoncev, S. N. On the localization of solutions of nonlinear degenerate elliptic and parabolic equations. Soviet Math. Dokl. 24 (1981), 420-424. · Zbl 0496.35051
[3] Aronson, D. G. Regularity properties of flows through porous media. SIAM J. Appl. Math. 17 (1969), 461-467. · Zbl 0187.03401
[4] Aronson, D. G. Regularity properties of flows through porous media: a counter-example. SIAM J. Appl. Math. 19 (1970), 299-307. · Zbl 0255.76099
[5] Aronson, D. G. Regularity properties of flows through porous media: the interface. Arch. Rational Mech. Anal. 37 (1970), 135-149. · Zbl 0202.37901
[6] Aronson, D. G. The porous medium equation, in Some Problems in Nonlinear Diffusion (A. Fasano & M. Primicerio, eds.). Lecture Notes in Math. 1224. Springer-Verlag, 1986. · Zbl 0626.76097
[7] Aronson, D. G., & Bénilan, Ph., Régularité des solutions de l’équation des milieux poreux dans 80-1. C. R. Acad. Sc. Paris 288 (1979), 103-105. · Zbl 0397.35034
[8] Aronson, D. G., Caffarelli, L. A., & Kamin, S., How an initially stationary interface begins to move in porous medium flow. SIAM J. Math. Anal. 14 (1983), 639-658. · Zbl 0542.76119
[9] Aronson, D. G., Caffarelli, L. A., & Vazquez, J. L., Interfaces with a corner point in one-dimensional porous medium flow. Comm. Pure Appl. Math. 38 (1985), 375-404. · Zbl 0571.35055
[10] Aronson, D. G., & Serrin, J., Local behavior of solutions of quasilinear parabolic differential equations. Arch. Rational Mech. Anal. 25 (1967), 81-123. · Zbl 0154.12001
[11] Aronson, D. G., & Vazquez, J. L., The porous medium equation, book in preparation for Pitman Monographs in Mathematics.
[12] Atkinson, C., & Bouillet, J., Some qualitative properties of solutions of a generalized diffusion equation. Math. Proc. Camb. Phil. Soc. 86 (1979), 495-510. · Zbl 0428.35046
[13] Aubin, J. P., Un théorème de compacité. C. R. Acad. Sc. Paris 256 (1963), 5042-5044. · Zbl 0195.13002
[14] Barenblatt, G. I., On self-similar motions of compressible fluids in porous media. Prikl. Mat. Mech. 16 (1952), 679-698. (Russian). · Zbl 0047.19204
[15] Bénilan, Ph., Sur un problème d’évolution non monotone dans L 2(?). Publ. Math. Fac. Sciences Besançon, N. 2 (1976).
[16] Bénilan, Ph., A strong regularity L pfor solutions of the porous media equation, in Contributions to Nonlinear P.D.E. (C. Bardos and others, eds.) Research Notes in Math. 89 (Pitman, 1983).
[17] Bénilan, Ph., & Crandall, M. G., Regularizing effects of homogeneous evolution equations, in Contributions to Analysis and Geometry, supplement to Amer. J. Math. (D. N. Clark and others, eds.), Baltimore (1981), 23-39.
[18] Berryman, J. G., & Holland, C. J., Stability of the separable solution for fast diffusion. Arch. Rational Mech. Anal. 74 (1980), 379-388. · Zbl 0458.35046
[19] Brézis, H., Monotonicity methods in Hubert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis. (E. Zarantonello, ed.). Academic Press (1971), 101-156.
[20] Caffarelli, L. A., & Friedman, A., Regularity of the free boundary for the one-dimensional flow of gas in a porous medium. Amer. J. Math. 101 (1979), 1193-1218. · Zbl 0439.76084
[21] van Duijn, H., & Hilhorst, D., On a doubly nonlinear diffusion equation in hydrology. Nonlinear Anal. T.M. & A., 11 (1987), 305-333. · Zbl 0654.35049
[22] Esteban, J. R., & Vazquez, J. L., On the equation of turbulent filtration in one-dimensional porous media. Nonlinear Anal. T.M. & A. 10 (1986), 1303-1325. · Zbl 0613.76102
[23] Gilding, B. G., & Peletier, L. A., The Cauchy problem for an equation in the theory of infiltration. Arch. Rational Mech. Anal. 61 (1976), 127-140. · Zbl 0336.76037
[24] Gurtin, M. E., & Maccamy, R. C., On the diffusion of biological populations. Math. Biosc. 33 (1977), 35-49. · Zbl 0362.92007
[25] Herrero, M. A., & Pierre, M., The Cauchy problem for ut=?(um) when 0<m<1. Trans. Amer. Math. Soc. 291 (1985), 145-158. · Zbl 0583.35052
[26] Herrero, M. A., & Vazquez, J. L., On the propagation properties of a nonlinear degenerate parabolic equation. Comm. PDE 7 (1982), 1381-1402. · Zbl 0516.35041
[27] Ilin, A. M., Kalashnikov, A. S., & Oleinik, O. A., Second-order linear equations of parabolic type. Russian Math. Surveys 17 (1962), 1-143. · Zbl 0108.28401
[28] Kalashnikov, A. S., On a nonlinear equation appearing in the theory of nonstationary filtration. Trud. Sem. I. G. Petrovsky 4 (1978), 137-146 (Russian). · Zbl 0415.35044
[29] Kalashnikov, A. S., On the Cauchy problem for second order degenerate parabolic equations with non power nonlinearities. Trud. Sem. I. G. Petrovsky 6 (1981), 83-96 (Russian). · Zbl 0461.35052
[30] Kalashnikov, A. S., On the propagation of perturbations in the first boundary-value problem for a doubly nonlinear degenerate parabolic equation. Trud. Sem. I. G. Petrovsky 8 (1982), 128-134 (Russian). · Zbl 0494.35010
[31] Knerr, B. F., The porous medium equation in one dimension. Trans. Amer. Math. Soc. 234 (1977), 381-415. · Zbl 0365.35030
[32] Kristianovitch, S. A., Motion of ground water which does not conform to Darcy’s law. Prikl. Mat. Mech. 4 (1940), 33-52 (Russian). · JFM 66.1113.08
[33] Ladyzenskaja, O. A., Solonnikov, V. A., & Uralceva, N. N., Linear and quasilinear equations of parabolic type. Translations of Math. Monographs, A.M.S. 1968.
[34] Leibenson, L. S., General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk SSSR, Geography and Geophysics 9 (1945), 7-10 (Russian).
[35] Martinson, L. K., & Pavlov, K. B., Unsteady shear flows of a conducting fluid with a rheological power law. Magnit. Gidrodinamika 2 (1971), 50-58.
[36] Muskat, M., The flow of homogeneous fluids through porous media. McGraw Hill, 1937. · JFM 63.1368.03
[37] Pavlov, K. B., & Romanov, A. S., Variation of the region of localization of disturbances in nonlinear transport processes. Izv. Akad. Nauk SSSR, Mech. Zhidk. Gaza, 6 (1980), 57-62 (Russian). · Zbl 0468.76052
[38] Peletier, L. A., The porous media equation, in Applications of Nonlinear Analysis in the Physical Sciences, (H. Amann and others, eds.). Pitman, 1981. · Zbl 0497.76083
[39] Raviart, P. A., Sur la résolution de certaines équations paraboliques non linéaires. J. Funct. Anal. 5 (1970), 299-328. · Zbl 0199.42401
[40] Vazquez, J. L., Asymptotic behavior and propagation properties of the one-dimensional flow of gas in a porous medium. Trans. Amer. Math. Soc. 277 (1983), 507-527. · Zbl 0528.76096
[41] Vazquez, J. L., The interface of one-dimensional flows in porous media. Trans. Amer. Math. Soc. 285 (1984), 717-737. · Zbl 0524.35060
[42] Vazquez, J. L., A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12 (1984), 191-202. · Zbl 0561.35003
[43] Volker, R. E., Nonlinear flow in porous media by finite elements. J. Hydraulics Div. Proc. Amer. Soc. Civil Eng. 95 (1969), 2093-2114.
[44] Widder, D. V., The heat equation. Academic Press, 1975. · Zbl 0322.35041
[45] Zeldovich, ya. B., & Raizer, Yu. P., Physics of shock waves and high temperature hydrodynamic phenomena. Academic Press, 1966.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.