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Harnack estimates and extinction profile for weak solutions of certain singular parabolic equations


Authors: E. DiBenedetto and Y. C. Kwong
Journal: Trans. Amer. Math. Soc. 330 (1992), 783-811
MSC: Primary 35B45; Secondary 35B05, 35K55, 35K65
DOI: https://doi.org/10.1090/S0002-9947-1992-1076615-7
MathSciNet review: 1076615
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Abstract: We establish an intrinsic Harnack estimate for nonnegative weak solutions of the singular equation $ (1.1)$ below, for $ m$ in the optimal range $ ((N - 2)_+/N,1)$. Intrinsic means that, due to the singularity, the space-time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself. Consequences are, sharp supestimates on the solutions and decay rates as $ t$ approaches the extinction time. Analogous results are shown for $ p$-laplacian type equations.


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  • [1] D. Andreucci and E. DiBenedetto, A new approach to initial traces in nonlinear filtration, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 305-334. MR 1067778 (92g:35089)
  • [2] D. G. Aronson and J. Serrin, Local behaviour of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal. 25 (1967), 81-123. MR 0244638 (39:5952)
  • [3] Ph. Bénilan and M. G. Crandall, The continuous dependence on $ \phi $ of solutions of $ {u_t} - \Delta \phi (u)= 0$, Indiana Univ. Math. J. 30 (1981), 161-177. MR 604277 (83d:35071)
  • [4] J. G. Berryman, Evolution of a stable profile for a class of nonlinear diffusion equations with fixed boundaries, J. Math. Phys. 18 (1977), 2108-2115.
  • [5] J. G. Berryman and C. J. Holland, Stability of the separable solution for fast diffusion equation, Arch. Rational Mech. Anal. 74 (1980), 379-388. MR 588035 (81m:35065)
  • [6] Chen Ya-zhe and E. DiBenedetto, On the local behaviour of solutions of singular parabolic equations, Arch. Rational Mech. Anal. 103 (1988), 319-345. MR 955531 (89k:35107)
  • [7] E. DiBenedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations, Arch. Rational Mech. Anal. 100 (1988), 129-147. MR 913961 (88j:35082)
  • [8] E. DiBenedetto and M. A. Herrero, Nonnegative solutions of the evolution $ p$-Laplacian equation. Initial traces and Cauchy problem when $ 1 < p < 2$, Arch. Rational Mech. Anal. 111 (1990), 225-290. MR 1066761 (92g:35088)
  • [9] J. Hadamard, Extension á l'équation de la chaleur d'un théorème de A. Harnack, Rend. Circ. Mat. Palermo 3 (1954), 337-346. MR 0068713 (16:930a)
  • [10] M. A. Herrero and M. Pierre, The Cauchy problem for $ {u_t} = \Delta ({u^m})$ when $ 0 < m < 1$, Trans. Amer. Math. Soc. 291 (1985), 145-158. MR 797051 (86i:35065)
  • [11] M. A. Herrero and J. L. Vazquez, Asymptotic behaviour of the solutions of a strongly non linear parabolic problem, Ann. Fac. Sci. Toulouse Math. 3 (1981), 113-127. MR 646311 (83e:35016)
  • [12] N. V. Krylov and M. V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR-Izv. 16 (1981), 151-164.
  • [13] Y. C. Kwong, Interior and boundary regularity of solutions to a plasma problem, Proc. Amer. Math. Soc. 104 (1988), 472-478. MR 962815 (90d:35143)
  • [14] -, Asymptotic behaviour of a plasma type equation with finite extinction, Arch. Rational Mech. Anal. 104 (1988), 277-294. MR 1017292 (90i:35042)
  • [15] -, Boundary behavior of solutions of the fast diffusion equation, Trans. Amer. Math. Soc. 322 (1990), 263-283. MR 1008697 (91b:35055)
  • [16] O. A. Ladyzhenskaya, N. A. Solonnikov, and N. N. Ural'tseva, Linear and quasilinear equations of parabolic type, Transl. Math. Monographs, vol. 23, Amer. Math. Soc., Providence, R. I., 1968.
  • [17] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134. MR 0159139 (28:2357)
  • [18] H. Okuda and J. M. Dawson, Numerical simulation on plasma diffusion in three dimensions, Phys. Lett. 28 (1972), 1625-1629.
  • [19] O. A. Oleinik, A. S. Kalashnikov, and Yui-Lin Chzhou, The Cauchy problem and boundary value problems for equations of the type of nonstationary filtration, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 667-704. (Russian) MR 0099834 (20:6271)
  • [20] M. Pierre, Nonlinear fast diffusion with measures as data, Proc. Nonlinear Parabolic Equations: Qualitative Properties of Solutions (Tesei and Boccardo, eds.), Pitman, New Yok, 1985.
  • [21] B. Pini, Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rend. Sem. Mat. Univ. Padova 23 (1954), 422-434. MR 0065794 (16:485c)
  • [22] J. Serrin, Local behaviour of solutions of quasilinear elliptic equations, Acta Math. 111 (1964), 101-134. MR 0170096 (30:337)
  • [23] N. S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205-226. MR 0226168 (37:1758)
  • [24] -, On Harnack type inequalities and their application to quasilinear elliptic partial differential equations, Comm. Pure Appl. Math. 20 (1967), 721-747. MR 0226198 (37:1788)

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DOI: https://doi.org/10.1090/S0002-9947-1992-1076615-7
Article copyright: © Copyright 1992 American Mathematical Society

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