×

Self-improving property of the fast diffusion equation. (English) Zbl 1423.35218

Summary: We show that the gradient of the \(m\)-power of a solution to a singular parabolic equation of porous medium-type (also known as fast diffusion equation), satisfies a reverse Hölder inequality in suitable intrinsic cylinders. Relying on an intrinsic Calderón-Zygmund covering argument, we are able to prove the local higher integrability of such a gradient for \(m \in(\frac{(n - 2)_+}{n + 2}, 1)\). Our estimates are satisfied for a general class of growth assumptions on the non linearity. In this way, we extend the theory for \(m \geq 1\) [the authors, Am. J. Math. 141, No. 2, 399–446 (2019; Zbl 1418.35057)] to the singular case. In particular, an intrinsic metric that depends on the solution itself is introduced for the singular regime.

MSC:

35K59 Quasilinear parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35K67 Singular parabolic equations

Citations:

Zbl 1418.35057
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Berryman, J. G., Evolution of a stable profile for a class of nonlinear diffusion equations with fixed boundaries, J. Math. Phys., 18, 11, 2108-2115 (1977) · Zbl 0403.76076
[2] Berryman, J. G.; Holland, C. J., Nonlinear diffusion problem arising in plasma physics, Phys. Rev. Lett., 40, 26, 1720-1722 (1978)
[3] Berryman, J. G.; Holland, C. J., Stability of the separable solution for fast diffusion equation, Arch. Ration. Mech. Anal., 74, 4, 379-388 (1980) · Zbl 0458.35046
[4] Bögelein, V., Higher integrability for weak solutions of higher order degenerate parabolic systems, Acad. Sci. Fenn. Math., 33, 2, 387-412 (2008) · Zbl 1154.35013
[5] Bögelein, V.; Duzaar, F.; Korte, R.; Scheven, C., The higher integrability of weak solutions of porous medium systems, Adv. Nonlinear Anal., 8, 1004-1034 (2019) · Zbl 1414.35111
[6] Bögelein, V.; Parviainen, M., Self-improving property of nonlinear higher order parabolic systems near the boundary, NoDEA Nonlinear Differential Equations Appl., 17, 21-54 (2010) · Zbl 1194.35087
[7] Carleman, T., Problèmes Matematiques dans la Théorie Cinétique des Gaz, Almqvist-Wiksells (1957), Almqvist-Wiksells · Zbl 0077.23401
[8] Chen, Y. Z.; DiBenedetto, E., Hölder estimates of solutions of singular parabolic equations with measurable coefficients, Arch. Ration. Mech. Anal., 118, 3, 257-271 (1992) · Zbl 0836.35029
[9] DiBenedetto, E., Degenerate Parabolic Equations, Universitext (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0794.35090
[10] DiBenedetto, E.; Friedman, A., Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357, 1-22 (1985) · Zbl 0549.35061
[11] DiBenedetto, E.; Gianazza, U.; Vespri, V., Harnack’s Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics (2012), Springer: Springer New York
[12] DiBenedetto, E.; Kwong, Y. C.; Vespri, V., Local space-analiticity of solutions of certain singular parabolic equations, Indiana Univ. Math. J., 40, 2, 741-765 (1991) · Zbl 0784.35055
[13] Diening, L.; Kaplický, P.; Schwarzacher, S., BMO estimates for the \(p\)-Laplacian, Nonlinear Anal., 75, 2, 637-650 (2012) · Zbl 1233.35056
[14] Evans, L. C.; Gariepy, R. F., Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics (1992), CRC Press: CRC Press Boca Raton, FL · Zbl 0804.28001
[15] Gehring, F. W., The \(L^p\)-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130, 265-277 (1973) · Zbl 0258.30021
[16] Gianazza, U.; Schwarzacher, S., Self-improving property of degenerate parabolic equations of porous medium-type, Amer. J. Math., 141, 2, 399-446 (2019) · Zbl 1418.35057
[17] Giaquinta, M.; Modica, G., Regularity results for some classes of higher order nonlinear elliptic systems, J. Reine Angew. Math., 311/312, 145-169 (1979) · Zbl 0409.35015
[18] Giaquinta, M.; Struwe, M., On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z., 179, 4, 437-451 (1982) · Zbl 0469.35028
[19] Giusti, E., Direct Methods in the Calculus of Variations (2003), World Scientific Publishing Co., Inc.: World Scientific Publishing Co., Inc. River Edge, NJ · Zbl 1028.49001
[20] Kinnunen, J.; Lewis, J. L., Higher integrability for parabolic systems of \(p\)-Laplacian type, Duke Math. J., 102, 2, 253-271 (2000) · Zbl 0994.35036
[21] Lions, P.-L.; Toscani, G., Diffusive limits for finite velocities Boltzmann kinetic models, Rev. Mat. Iberoam., 13, 473-513 (1997) · Zbl 0896.35109
[22] Meyers, N. G.; Elcrat, A., Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions, Duke Math. J., 42, 121-136 (1975) · Zbl 0347.35039
[23] Parviainen, M., Global gradient estimates for degenerate parabolic equations in nonsmooth domains, Ann. Mat. Pura Appl. (4), 188, 2, 333-358 (2009) · Zbl 1179.35080
[24] Parviainen, M., Reverse Hölder inequalities for singular parabolic equations near the boundary, J. Differential Equations, 246, 512-540 (2009) · Zbl 1173.35037
[25] Schwarzacher, S., Hölder-Zygmund estimates for degenerate parabolic systems, J. Differential Equations, 256, 7, 2423-2448 (2014) · Zbl 1288.35284
[26] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43 (1993), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0821.42001
[27] Stredulinsky, E. W., Higher integrability from reverse Hölder inequalities, Indiana Univ. Math. J., 29, 3, 407-413 (1980) · Zbl 0442.35064
[28] Vázquez, J. L., Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Series in Mathematics and Its Applications, vol. 33 (2006), Oxford University Press: Oxford University Press Oxford · Zbl 1113.35004
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.