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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Cauchy problem for $ u\sb t=\Delta u\sp m$ when $ 0<m<1$


Authors: Miguel A. Herrero and Michel Pierre
Journal: Trans. Amer. Math. Soc. 291 (1985), 145-158
MSC: Primary 35K55; Secondary 76X05
DOI: https://doi.org/10.1090/S0002-9947-1985-0797051-0
MathSciNet review: 797051
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Abstract: This paper deals with the Cauchy problem for the nonlinear diffusion equation $ \partial u/\partial t - \Delta (u\vert u{\vert^{m - 1}}) = 0$ on $ (0,\infty ) \times {{\mathbf{R}}^N},u(0, \cdot ) = {u_0}$ when $ 0 < m < 1$ (fast diffusion case). We prove that there exists a global time solution for any locally integrable function $ {u_0}$: hence, no growth condition at infinity for $ {u_0}$ is required. Moreover the solution is shown to be unique in that class. Behavior at infinity of the solution and $ L_{\operatorname{loc} }^\infty $-regularizing effects are also examined when $ m \in (\max \{ (N - 2)/N,0\} ,1)$.


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  • [1] D. G. Aronson and Ph. Bënilan, Régularité des solutions de l'équation des milieux poreux dans $ {{\mathbf{R}}^N}$, C. R. Acad. Sci. Paris Sér. A 288 (1979), 103-105.
  • [2] D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1983), 351-366. MR 712265 (85c:35042)
  • [3] P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble) 34 (1984), 185-206. MR 743627 (86j:35063)
  • [4] -, Problèmes paraboliques semi-linéaires avec donnés mesures, Applicable Anal. 18 (1984), 111-149. MR 762868 (87k:35116)
  • [5] G. I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Mekh. 16 (1952), 67-78. (Russian) MR 0046217 (13:700a)
  • [6] Ph. Bénilan and M. G. Crandall, Regularizing effects of homogeneous evolution equations, Contributions to Analysis and Geometry (D. N. Clark et al., eds.), John Hopkins Univ. Press, Baltimore, Md., 1981, pp. 23-30. MR 648452 (83g:47063)
  • [7] H. Brezis, Semilinear equations in $ {{\mathbf{R}}^N}$ without conditions at infinity (to appear).
  • [8] Ph. Bénilan, M. G. Crandall and M. Pierre, Solutions of the porous medium equation in $ {{\mathbf{R}}^N}$ under optimal conditions on initial values, Indiana Univ. Math. J. 33 (1984), 51-87. MR 726106 (86b:35084)
  • [9] H. Brezis and M. G. Crandall, Uniqueness of solutions of the initial value problem for $ {u_t} - \Delta \varphi (u) = 0$, J. Math. Pures Appl. 56 (1979), 153-163. MR 539218 (80e:35029)
  • [10] B. J. Dahlberg and C. E. Kenig, Non-negative solutions of the porous medium equation, Comm. Partial Differential Equations 9 (1984), 409-438.
  • [11] E. Di Benedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J. 32 (1983), 83-118. MR 684758 (85c:35010)
  • [12] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, N.J., 1964. MR 0181836 (31:6062)
  • [13] T. Kato, Schrödinger operators with singer potentials, Israel J. Math. 13 (1972), 135-148. MR 0333833 (48:12155)
  • [14] O. A. Oleinik, A. S. Kalashnikov and C. Yu Lin, The Cauchy problem and boundary problems for equations of the type of unsteady filtration, Izv. Akad. Nauk SSR Ser. Mat. 22 (1958), 667-704. MR 0099834 (20:6271)
  • [15] L. A. Peletier, The porous medium equation, Application of Non-linear Analysis in the Physical Sciences (H. Amann et al., eds.), Pitman, London, 1981, pp. 229-241. MR 659697 (83k:76076)
  • [16] P. Sacks, Continuity of solutions of degenerate parabolic equations, Thesis, Univ. of Wisconsin, Madison, 1981.
  • [17] J. L. Vazquez, Behaviour of the velocity of one dimensional flows in porous media, Trans. Amer. Math. Soc. 286 (1984), 787-802. MR 760987 (86b:35016)
  • [18] L. Veron, Effets régularisants de semi-groupes non-linéaires dans des espaces de Banach, Ann. Fac. Sci. Toulouse Math. (5) 1 (1979), 171-200. MR 554377 (81i:47066)
  • [19] N. A. Watson, The rate of spatial decay of non-negative solutions of linear parabolic equations, Arch. Rational Mech. Anal. 68 (1978), 121-125. MR 505509 (80b:35023)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0797051-0
Keywords: Cauchy problem, nonlinear diffusion, initial-value problem, regularizing effects
Article copyright: © Copyright 1985 American Mathematical Society

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