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Local vanishing properties of solutions of elliptic and parabolic quasilinear equations

Authors: J. Ildefonso Díaz and Laurent Véron
Journal: Trans. Amer. Math. Soc. 290 (1985), 787-814
MSC: Primary 35B05; Secondary 35J60, 35K55
MathSciNet review: 792828
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Abstract: We use a local energy method to study the vanishing property of the weak solutions of the elliptic equation $ - \operatorname{div}\;A(x,u,Du) + B(x,u,Du) = 0$ and of the parabolic equation $ \partial \psi (u)/\partial t - \operatorname{div}\;\mathcal{A}(t,x,u,Du) + \mathcal{B}(t,x,u,Du) = 0$. The results are obtained without any assumption of monotonicity on $ A$, $ B$, $ \mathcal{A}$ and $ \mathcal{B}$.

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  • [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [2] Hans Wilhelm Alt and Stephan Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z. 183 (1983), no. 3, 311–341. MR 706391, 10.1007/BF01176474
  • [3] S. N. Antoncev, On the localization of solutions of nonlinear degenerate elliptic and parabolic equations, Soviet Math. Dokl. 24 (1981), 420-424.
  • [4] R. Aris, The mathematical theory of diffusion and reaction in permeable catalysts, Clarendon Press, Oxford, 1975.
  • [5] C. Atkinson and J. E. Bouillet, Some qualitative properties of solutions of a generalised diffusion equation, Math. Proc. Cambridge Philos. Soc. 86 (1979), no. 3, 495–510. MR 542697, 10.1017/S030500410005636X
  • [6] H. Attouch and A. Damlamian, Application des méthodes de convexité et monotonie à l’étude de certaines équations quasi linéaires, Proc. Roy. Soc. Edinburgh Sect. A 79 (1977/78), no. 1-2, 107–129 (French, with English summary). MR 0477473
  • [7] Alain Bamberger, Étude d’une équation doublement non linéaire, J. Functional Analysis 24 (1977), no. 2, 148–155 (French). MR 0470490
  • [8] Philippe Benilan, Haim Brezis, and Michael G. Crandall, A semilinear equation in 𝐿¹(𝑅^{𝑁}), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4, 523–555. MR 0390473
  • [9] Marie-Françoise Bidaut-Véron, Variational inequalities of order 2𝑚 in unbounded domains, Nonlinear Anal. 6 (1982), no. 3, 253–269. MR 654317, 10.1016/0362-546X(82)90093-1
  • [10] Haïm Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 101–156. MR 0394323
  • [11] H. Brezis, Solutions of variational inequalities, with compact support, Uspehi Mat. Nauk 29 (1974), no. 2 (176), 103–108 (Russian). Translated from the English by Ju. A. Dubinskiĭ; Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. MR 0481460
  • [12] H. Brézis and F. E. Browder, Strongly nonlinear elliptic boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 3, 587–603. MR 507004
  • [13] Haïm Brézis and Avner Friedman, Estimates on the support of solutions of parabolic variational inequalities, Illinois J. Math. 20 (1976), no. 1, 82–97. MR 0390501
  • [14] J. Ildefonso Díaz Díaz, Solutions with compact support for some degenerate parabolic problems, Nonlinear Anal. 3 (1979), no. 6, 831–847. MR 548955, 10.1016/0362-546X(79)90051-8
  • [15] Ildefonso Díaz and Jesús Hernández, Some results on the existence of free boundaries for parabolic reaction-diffusion systems, Trends in theory and practice of nonlinear differential equations (Arlington, Tex., 1982) Lecture Notes in Pure and Appl. Math., vol. 90, Dekker, New York, 1984, pp. 149–156. MR 741498
  • [16] J. I. Díaz and M. A. Herrero, Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems, Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), no. 3-4, 249–258. MR 635761, 10.1017/S0308210500020266
  • [17] J. Ildefonso Diaz and Laurent Véron, Existence theory and qualitative properties of the solutions of some first order quasilinear variational inequalities, Indiana Univ. Math. J. 32 (1983), no. 3, 319–361. MR 697641, 10.1512/iumj.1983.32.32025
  • [18] Morton E. Gurtin and Richard C. MacCamy, On the diffusion of biological populations, Math. Biosci. 33 (1977), no. 1-2, 35–49. MR 0682594
  • [19] A. S. Kalashnikov, On equations of the nonstationary-filtration type in which the perturbation is propagated at infinite velocity, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1972), 45-49.
  • [20] A. S. Kalašnikov, The nature of the propagation of perturbations in problems of nonlinear heat conduction with absorption, Ž. Vyčisl. Mat. i Mat. Fiz. 14 (1974), 891–905, 1075 (Russian). MR 0352702
  • [21] -, On a nonlinear equation appearing in the theory of non-stationary filtration, Trudy Sem. Petrovsk. 4 (1978), 137-146.
  • [22] R. Kershner, The behavior of temperature fronts in media with nonlinear heat conductivity under absorption, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 (1978), 44–51 (Russian, with English summary). MR 516020
  • [23] -, Filtration with absorption: necessary and sufficient conditions for the propagation of perturbations to have finite velocity (to appear).
  • [24] -, Nonlinear heat conduction with absorption: space localization and extinction in finite time (to appear).
  • [25] Barry F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc. 234 (1977), no. 2, 381–415. MR 0492856, 10.1090/S0002-9947-1977-0492856-3
  • [26] Olga A. Ladyzhenskaya and Nina N. Ural′tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR 0244627
  • [27] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and quasilinear equations of parabolic type, Amer. Math. Soc. Transl., vol. 23, 1968.
  • [28] J. L. Lions, Quelques méthodes de résolutions des problèmes aux limites non linéaires, Dunod, Paris, 1969.
  • [29] L. K. Martinson and K. B. Pavlov, The effect of magnetic plasticity in non-Newtonian fluids, Magnit. Gidrodinamika 3 (1969), 69-75.
  • [30] -, Unsteady shear flows of a conducting fluid with a rheological power law, Magnit. Gidrodinamika 4 (1970), 50-58.
  • [31] O. A. Oleĭnik, A. S. Kalašinkov, and Yuĭ-Lin′ Čžou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR. Ser. Mat. 22 (1958), 667–704 (Russian). MR 0099834
  • [32] L. A. Peletier, A necessary and sufficient condition for the existence of an interface in flows through porous media, Arch. Rational Mech. Anal. 56 (1974/75), 183–190. MR 0417572
  • [33] Guido Stampacchia, Èquations elliptiques du second ordre à coefficients discontinus, Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965), Les Presses de l’Université de Montréal, Montreal, Que., 1966 (French). MR 0251373
  • [34] Laurent Véron, Équations d’évolution semi-linéaires du second ordre dans 𝐿¹, Rev. Roumaine Math. Pures Appl. 27 (1982), no. 1, 95–123 (French). MR 658869
  • [35] Laurent Véron, Effets régularisants de semi-groupes non linéaires dans des espaces de Banach, Ann. Fac. Sci. Toulouse Math. (5) 1 (1979), no. 2, 171–200 (French, with English summary). MR 554377

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Keywords: Elliptic equations, parabolic equations, weak solutions, local energy estimates, differential inequalities, trace-interpolation estimates, free boundary, propagation phenomena
Article copyright: © Copyright 1985 American Mathematical Society