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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 ${u}_{t}=\nabla ·\left(|\nabla {u}^{m}{|}^{p-2}\nabla {u}^{m}\right)-{u}^{q}$ in ${ℝ}^{N}×\left(0,\infty \right)$, $u\left(·,0\right)=\mu$ in ${ℝ}^{N}$, where $p>1$, $m>0$, $q>0$ and $\mu$ is a nonnegative bounded Radon measure. The existence is proved provided $m\left(p-1\right)+p/N>1$ and $q. It is also shown that these conditions are essentially optimal. The existence proof is based on the approximation of $\mu$ by smooth functions.
##### MSC:
 35K65 Parabolic equations of degenerate type 35R05 PDEs with discontinuous coefficients or data 35K15 Second 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 $ℝ$ 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)