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Degenerate parabolic equations and Harnack inequality. (English) Zbl 0573.35052
The author constructs examples to show that, in general, solutions of uniformly parabolic diagonal systems with quadratic growth and smooth initial and boundary data may lose the regularity properties of the initial data in finite time. He raises the question as to whether regularity is preserved when the parabolic systems are of variational type.
Reviewer: D.T.Haimo

MSC:
 35K65 Degenerate parabolic equations 35B65 Smoothness and regularity of solutions to PDEs 35K20 Initial-boundary value problems for second-order parabolic equations
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References:
 [1] Coifman, R.; Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals, Studia Math., LI, 241-250 (1974) · Zbl 0291.44007 [2] E.Fabes - C.Kenig - R.Serapioni,The local regularity of solutions of degenerate elliptic equations, Comm. Partial Diff. Equations,7 (1) (1982). · Zbl 0498.35042 [3] Ivanov, A. V., Smoothness of generalized solutions of degenerate parabolic equations of second order, Proc. Steklov Inst. Math., 116, 52-67 (1971) [4] Ivanov, A. V., The boundary-value problem for linear parabolic equations of the divergence type with measurable coefficients, J. Soviet Math., 9, 651-680 (1978) · Zbl 0396.35054 [5] Ivanov, A. V., Properties of solutions of linear and quasilinear second order equations with measurable coefficients which are neither strictly nor uniformly parabolic, J. Soviet Math., 10, 29-43 (1978) · Zbl 0388.35038 [6] S.Kruzhkov - I.Kolodii,A priori estimates and Harnack’s inequality for generalized solutions of degenerate quasi linear second order parabolic equations, Soviet Math. Dokl., Vol.13, N∘3 (1972). [7] L. J.Lions,Equations différentielles opérationnelles et problèmes aux limites, Springer Verlag, 1961. · Zbl 0098.31101 [8] Ladyzenskaja, O.; Solonnikov, V.; Ural’Ceva, N., Linear and quasilinear equations of parabolic type (1968), Providence: Amer. Math. Soc., Providence [9] Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165, 207-226 (1972) · Zbl 0236.26016 [10] Murthy, M. K. V.; Stampacchia, G., Boundary value problems for some degenerate elliptic operators, Ann. Mat. Pura Appl., (4), 20, 1-122 (1968) · Zbl 0185.19201 [11] Murthy, M. K. V.; Stampacchia, G., Errata Corrige, Ann. Mat. Pura Appl., (4), 90, 413-414 (1971) · Zbl 0226.35037 [12] Muckenhoupt, B.; Wheeden, R., Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 198, 361-274 (1974) · Zbl 0289.26010 [13] Nicolosi, F., Soluzioni deboli dei problemi al eontorno per operatori parabolici che possono degenerare, Ann. Mat. Pura Appl., (4), 185, 135-155 (1980) · Zbl 0452.35065 [14] F.Nicolosi,Sulla limitatezza delle soluzioni deboli dei problemi al contorno per operatori parabolici degeneri, to appear in Rendiconti Circ. Mat. Palermo. · Zbl 0507.35046 [15] G.Stampacchia,Equations elliptiques du second ordre à coefficients discontinus, Montreal, 1966. · Zbl 0151.15501 [16] E.Stein,Singular integrals and differentiability properties of functions, Princeton University Press, 1970. · Zbl 0207.13501 [17] Treves, F., Basic linear partial differential equations (1975), New York: Academic Press, New York · Zbl 0305.35001 [18] Trudinger, N., On the regularity of generalized solutions of linear, non-uniformly elliptic equations, Arch. Rat. Mech. Anal., 48, 51-62 (1971) · Zbl 0218.35035 [19] Trudinger, N., Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Super. Pisa, 87, 265-308 (1973) · Zbl 0279.35025 [20] Trudinger, N., Generalized solutions of quasilinear differential inequalities. I. Elliptic operators, Bull. Amer. Math. Soc., 77, 576-579 (1971) · Zbl 0235.35013
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