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Cauchy problem of some doubly degenerate parabolic equations with initial datum a measure. (English) Zbl 1073.35130
The author studies the existence of nonnegative solutions of the Cauchy problem u t =·(|u m | p-2 u m )-u q in N ×(0,), u(·,0)=μ in N , where p>1, m>0, q>0 and μ is a nonnegative bounded Radon measure. The existence is proved provided m(p-1)+p/N>1 and q<m(p-1)+p/N. It is also shown that these conditions are essentially optimal. The existence proof is based on the approximation of μ by smooth functions.
MSC:
35K65Parabolic equations of degenerate type
35R05PDEs with discontinuous coefficients or data
35K15Second order parabolic equations, initial value problems
References:
[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) · Zbl 0613.76102 · doi:10.1016/0362-546X(86)90068-4
[2]Esteban, J. A., Vazquez, J. L.: Homogeneous diffusion in with powerlike nonlinear diffusivity. Arch. Rat. Mech. Anal., 103 (1), 39–80 (1988) · Zbl 0683.76073 · doi:10.1007/BF00292920
[3]Tsutsumi, M.: on solution of some doubly nonlinear degenerate parabolic equation with absorption. J. Math. Anal. and Appl., 132, 187–212 (1988) · Zbl 0681.35047 · doi:10.1016/0022-247X(88)90053-4
[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)
[5]Ivanov, A. V.: Hölder continuity of solutions for nonlinear degenerate parabolic equations. J. Soviet. Math., 56(2), 2320–2347 (1991) · Zbl 0729.35018 · doi:10.1007/BF01671935
[6]Yuan, H. J.: Hölder continuity of solutions for nonlinear degenerate parabolic equations. Acta. Sci. Nat. Jilin., (2) (1991)
[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)
[8]Kersner, R.: Degenerate Parabolic Equations with General Nonlinearities, Nonlinear Analysis, Theory, Methods and Applications, 4(6), 1043–1062 (1980) · Zbl 0452.35062 · doi:10.1016/0362-546X(80)90015-2
[9]Ladyzenskaya, O. A., Solonikov, V. A., Uratceva, N. N.: Linear and quasilinear equations of parabolic type. Amer. Math. Soc., Providence (1968)