A Notion of Rectifiability Modeled on Carnot Groups

Nicholas Alikakos, Rouben Rostamian, Large Time Behavior of Solutions of Neumann Boundary Value Problem for the Porous Medium Equation, Indiana Univ. Math. J. 30 (1981), 749-785

Abstract

References

[1] N. D. Alikakos, $L\sp{p}$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4, No. 8 (1979), pp. 827–868. MathSciNet. Zentralblatt für Mathematik.

[2] Nicholas D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33, No. 2 (1979), pp. 201–225. MathSciNet. Zentralblatt für Mathematik.

[3] D. G. Aronson, Regularity properties of flows through porous media: A counterexample, SIAM J. Appl. Math., 19 (1970), pp. 299–307. MathSciNet. Zentralblatt für Mathematik.

[4] D. G. Aronson, Regularity propeties of flows through porous media, SIAM J. Appl. Math., 17 (1969), pp. 461–467. MathSciNet. Zentralblatt für Mathematik.

[5] Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest, 1976, pp. 352. MathSciNet. Zentralblatt für Mathematik.

[6] James G. Berryman and Charles J. Holland, Stability of the separable solution for fast diffusion, Arch. Rational Mech. Anal., 74, No. 4 (1980), pp. 379–388. CrossRef. MathSciNet. Zentralblatt für Mathematik.

[7] Ph. Benilan and M. G. Crandall, The continuous dependence of solutions of $u_t-\Delta \phi(u)= 0$ , MRC Tech. Summ. Rep., #1942, Univ. of Wisconsin, 1979.

[8] H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L\sp{1}$ , J. Math. Soc. Japan, 25 (1973), pp. 565–590. MathSciNet. Zentralblatt für Mathematik.

[9] David C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22 (1972/73), pp. 65–74. MathSciNet. Zentralblatt für Mathematik.

[10] Michael G. Crandall, An introduction to evolution governed by accretive operators, Dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., 1974), Vol. I, Cesari, ed., Academic Press, New York, 1976, pp. 131–165. MathSciNet. Zentralblatt für Mathematik.

[11] M. G. Crandall and T. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), pp. 265–298. MathSciNet.

[12] C. M. Dafermos, Asymptotic behavior of solutions of evolution equations, Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), M. G. Crandall, ed., Publ. Math. Res. Center Univ. Wisconsin, vol. 40, Academic Press, New York, 1978, pp. 103–123. MathSciNet. Zentralblatt für Mathematik.

[13] C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Functional Analysis, 13 (1973), pp. 97–106. CrossRef. MathSciNet. Zentralblatt für Mathematik.

[14] Lawrence C. Evans, Application of nonlinear semigroup theory to certain partial differential equations, Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), M. G. Crandall, ed., Publ. Math. Res. Center Univ. Wisconsin, vol. 40, Academic Press, New York, 1978, pp. 163–188. MathSciNet. Zentralblatt für Mathematik.

[15] Morton E. Gurtin and Richard C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33, No. 1-2 (1977), pp. 35–49. CrossRef. MathSciNet. Zentralblatt für Mathematik.

[16] Jack K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), pp. 39–59. CrossRef. MathSciNet. Zentralblatt für Mathematik.

[17] A. S. Kalashnikov, Formation of singularities in solutions of the equation of nonstationary filtration, ū Z. Vyčisl. Mat. i Mat. Fiz., 7 (1967), pp. 440–444. MathSciNet.

[18] Yoshio Konishi, On the nonlinear semi-groups associated with $u\sb{t}=\Delta \beta (u)$ and $\varphi (u\sb{t})=\Delta u$ , J. Math. Soc. Japan, 25 (1973), pp. 622–628. MathSciNet. Zentralblatt für Mathematik.

[19] O. A. Ladyzenskaja, V. A. Sollonnikov, and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967, English Transl., Transl. Math. Monographs, Vol. 23, AMS, Providence, R.I., 1968.

[20] J. P. LaSalle, Asymptotic stability criteria, Proc. Sympos. Appl. Math., Vol. XIII, American Mathematical Society, Providence, R.I., 1962, pp. 299–307. MathSciNet. Zentralblatt für Mathematik.

[21] O. A. Oleinik, On some degenerate quasilinear parabolic equations, Instituto Nazionale di Alta Matematica Seminari 1962-1963, Cremonese, Rome, 1965.

[22] O. A. Oleinik, A. S. Kalashnikov, and Chzou Yui-Lin, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR. Ser. Mat., 22 (1958), pp. 667–704. MathSciNet.

[23] A. Pazy, Semigroups of nonlinear contractions and their asymptotic behaviour, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. III (Heriot-Watt Univ., Edinburgh), R. J. Knops, ed., Res. Notes in Math., vol. 30, Pitman, Boston, Mass., 1979, pp. 36–134. MathSciNet.

[24] P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, Course of lectures, CIME, Varenna, Italy, 1974.

[25] L. A. Caffarelli and L. C. Evans, Continuity of the temperature in the two-phase Stefan problem, Arch. Rational Mech. Anal., 81, No. 3 (1983), pp. 199–220. CrossRef. MathSciNet. Zentralblatt für Mathematik.

[26] N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience, New York, 1957.

[27] S. N. Kruzhkov, Results on the nature of the continuity of solutions of parabolic equations, and certain applications thereof, Mat. Zametki, 6 (1969), pp. 97–108, Translated as Math. Notes 6 (1969), 517–523. MathSciNet.

[28] B. H. Gilding, Hölder continuity of solutions of parabolic equations, J. London Math. Soc. (2), 13, No. 1 (1976), pp. 103–106. MathSciNet. Zentralblatt für Mathematik.

[29] Nicholas D. Alikakos and Rouben Rostamian, Lower bound estimates and separable solutions for homogeneous equations of evolution in Banach space, J. Differential Equations, 43, No. 3 (1982), pp. 323–344. CrossRef. MathSciNet. Zentralblatt für Mathematik.

[30] H. Brezis and M. G. Crandall, Uniqueness of solutions of the initial-value problem for $u\sb{t}-\Delta \varphi (u)=0$ , J. Math. Pures Appl. (9), 58, No. 2 (1979), pp. 153–163. MathSciNet. Zentralblatt für Mathematik.

[31] Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York, 1969, p. vi+262. MathSciNet. Zentralblatt für Mathematik.

[32] D. G. Aronson and L. A. Peletier, Large time behaviour of solutions of the porous medium equation in bounded domains, J. Differential Equations, 39, No. 3 (1981), pp. 378–412. MathSciNet. Zentralblatt für Mathematik.

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