Acta Mathematica Sinica

, Volume 20, Issue 4, pp 663–682

Cauchy Problem of Some Doubly Degenerate Parabolic Equations with Initial Datum a Measure

ORIGINAL ARTICLES

Abstract

This paper discusses the Cauchy problem of the equation
$$ u_{t} = \nabla \cdot {\left( {{\left| \nabla \right|}_{u} ^{m} \left| {^{{p - 2}} } \right.\nabla _{u} ^{m} } \right)} - u^{q} $$
(1)
with initial datum a measure. Under the assumption of the parameters, one proves the existence and non-existence of the non-negative generalized solution.

Keywords

Cauchy problem Degenerate Existence Measure 

MR (2000) Subject Classification

35K65 

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References

  1. 1.
    Esteban, J. A., Vazquez, J. L.: On the equation of tirbulent filtration in one dimensional porous media. Nonlinear. Anal. T. M. A., 10, 1303–1325 (1986)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Esteban, J. A., Vazquez, J. L.: Homogeneous diffusion in ℝ with powerlike nonlinear diffusivity. Arch. Rat. Mech. Anal., 103 (1), 39–80 (1988)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Tsutsumi, M.: on solution of some doubly nonlinear degenerate parabolic equation with absorption. J. Math. Anal. and Appl., 132, 187–212 (1988)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Zhao, J. N., Yuan, H. J.: The Cauchy problem of some doubly nonlinear degenerate parabolic equations. Chinese Ann. Math., 16A(2), 181–196 (1995)Google Scholar
  5. 5.
    Ivanov, A. V.: Hölder continuity of solutions for nonlinear degenerate parabolic equations. J. Soviet. Math., 56(2), 2320–2347 (1991)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Yuan, H. J.: Hölder continuity of solutions for nonlinear degenerate parabolic equations. Acta. Sci. Nat. Jilin., (2) (1991)Google Scholar
  7. 7.
    Oleinik O. A., Kalashnikov A. S., Chzou, Y. L.: The Cauchy problem and boundary value problems for equations of the type of nonstationary filtration. Izv. Akad. Nauk. SSSR. Soc. Mat., 22 (1958)Google Scholar
  8. 8.
    Kersner, R.: Degenerate Parabolic Equations with General Nonlinearities, Nonlinear Analysis, Theory, Methods and Applications, 4(6), 1043–1062 (1980)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Ladyzenskaya, O. A., Solonikov, V. A., Uratceva, N. N.: Linear and quasilinear equations of parabolic type. Amer. Math. Soc., Providence (1968)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institute of MathematicsAcademy of Mathematics and System Sciences, Chinese Academy of SciencesBeijing 100080P. R. China

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