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Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations. (English) Zbl 0708.35017
The author considers degenerate parabolic equations of the type \[ (1)\quad u_ t-div(| \nabla u|^{p-2}\nabla u)=0,\quad p>2,\text{ in } D'(\Omega_ T);\quad u\in C(0,T;L^ 2(\Omega))\cap L^ p(0,T;W^{1,p}(\Omega)) \] and of the type (2) \(u_ t-\Delta u^{m=0}\), \(m>1\), in \(D'(\Omega_ T)\), \(u\in C(0,T;L^ 2(\Omega))\), \(u^ m\in L^ 2(0,T;W^{1,2}(\Omega))\), where \(\Omega\) is an open set of \(R^ N\), \(0<T<\infty\), \(\Omega_ T\equiv \Omega \times (0,T]\). For nonnegative weak solutions of the equation (1) the author gives an intrinsic Harnack type inequality: \[ u(x_ 0,t_ 0)\leq C_ 0\inf_{x\in B_ R(x_ 0)}u(x,t_ 0+\theta) \] (if \(u(x_ 0,t_ 0)>0\), \(\theta =C_ 1R^ p/[u(x_ 0,t_ 0)]^{p-2}\) and \(B_{2R}(x_ 0)\times (t_ 0-\theta,t+\theta)\subset \Omega_ T)\) etc. For the equation (2), an analogous result is obtained.
Reviewer: Zheng Xingli

35B45 A priori estimates in context of PDEs
35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
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[1] D. G. Aronson & L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. AMS 280 (1983) # 1 pp. 351–366. · Zbl 0556.76084 · doi:10.1090/S0002-9947-1983-0712265-1
[2] D. G. Aronson & J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal. Vol. 25 1967 pp. 81–123. · Zbl 0154.12001 · doi:10.1007/BF00281291
[3] G. I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Meh. 16 (1952) pp. 67–78. · Zbl 0049.41902
[4] M. Bertsch & L. A. Peletier, A positivity property of solutions of non-linear diffusion equations, Journ. of Diff. Equ. 53 (1984) # 1 pp. 30–47. · Zbl 0531.35049 · doi:10.1016/0022-0396(84)90024-X
[5] B. E. Dahlberg & C. E. Kenig, Non linear filtration (to appear). · Zbl 0644.35057
[6] E. Di Benedetto & A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, Journ. für die Reine und Angew. Math. 357 (1985) pp. 1–22.
[7] E. Di Benedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Sc. Norm. Sup. Pisa Cl. Scienze, Serie IV, Vol. XIII, n. 3 (1986) pp. 487–535.
[8] J. Hadamard, Extension à l’équation de la chaleur d’un théorème de A. Harnack, Rend. Circ. Mat. Palermo Ser. 2 Vol. 3 (1954) pp. 337–346. · Zbl 0058.32201 · doi:10.1007/BF02849264
[9] N. V. Krylov & M. V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR Izvestijia Vol. 16 (1981) # 1 pp. 151–164. · Zbl 0464.35035 · doi:10.1070/IM1981v016n01ABEH001283
[10] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. Vol. 17 (1964) pp. 101–134. · Zbl 0149.06902 · doi:10.1002/cpa.3160170106
[11] J. Serrin, Local behavior of solutions of quasilinear elliptic equations, Acta Math. Vol. 111 (1964) pp. 302–347. · Zbl 0128.09101 · doi:10.1007/BF02391014
[12] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic partial differential equations, Comm. Pure Appl. Math. Vol. 20 (1967) pp. 721–747. · Zbl 0153.42703 · doi:10.1002/cpa.3160200406
[13] N. S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. XXI (1968) pp. 205–226. · Zbl 0159.39303 · doi:10.1002/cpa.3160210302
[14] E. Di Benedetto & M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation (to appear).
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