Struwe, Michael A counterexample in regularity theory for parabolic systems. (English) Zbl 0573.35053 Czech. Math. J. 34(109), 183-188 (1984). The authors study the degenerate parabolic partial differential equation \[ -(\partial /\partial x_ i)(a_{ij}(x,t)(\partial u/\partial x_ j))+(\partial u/\partial t)=f \] under suitable assumptions on the coefficients, deriving global properties of the solutions, including the \(L^ 2\) continuity of solutions to the Cauchy-Dirichlet problem. They construct examples to show that, when \(f=0\), the equation fails to have local \(L^{\infty}\) estimates unlike the elliptic case. Other examples are provided to establish that Harnack’s inequality cannot hold. Reviewer: O.T.Haimo Cited in 1 ReviewCited in 8 Documents MSC: 35K65 Degenerate parabolic equations 35B65 Smoothness and regularity of solutions to PDEs 35D10 Regularity of generalized solutions of PDE (MSC2000) 35K20 Initial-boundary value problems for second-order parabolic equations Keywords:continuity; Cauchy-Dirichlet problem; local \(L^{\infty }\) estimates; Harnack’s inequality PDFBibTeX XMLCite \textit{M. Struwe}, Czech. Math. J. 34(109), 183--188 (1984; Zbl 0573.35053) Full Text: EuDML References: [1] Frehse J.: A discontinuous solution of a mildly nonlinear elliptic system. Math. Z. 134, 229-230 (1973). · Zbl 0267.35038 · doi:10.1007/BF01214096 [2] Giaquinta M., M. Struwe: An optimal regularity result for quasilinear parabolic systems. manusc. math. 36, 223-239 (1981). · Zbl 0475.35026 · doi:10.1007/BF01170135 [3] Heinz E.: On certain nonlinear elliptic differential equations and univalent mappings. J. Analyse math. 5, 197-272 (1956/57). · Zbl 0085.08701 · doi:10.1007/BF02937346 [4] Hildebrandt S., K.-O. Widman: Some regularity results for quasilinear elliptic systems of second order. Math. Z. 142, 67-86 (1975). · Zbl 0317.35040 · doi:10.1007/BF01214849 [5] Ladyshenskaya O. A. V. A. Solonnikov, N. N. Ural’ceva: Linear and quasilinear equations of parabolic type. Transl. Math. Monographs 23, AMS, Providence, R. I. (1968). [6] Sampson J. H.: On the heat equation for harmonic maps. preprint. · Zbl 0497.53050 [7] Struwe M.: On the Holder continuity of bounded weak solutions of quasilinear parabolic systems. Manusc. math. 35, 125-145 (1981). · Zbl 0519.35007 · doi:10.1007/BF01168452 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.