Local Hölder Regularity of Weak Solutions for Singular Parabolic Systems of p-Laplacian Type

Khoirunisa Khoirunisa, Corina Karim, M. Muslikh

Abstract


In this paper, the Hölder regularity of weak solutions for singular parabolic systems of p-Laplacian type is investigated. By the Poincare inequality, we show that its weak solutions within Hölder space.


Keywords


Holder regularity, singular case, weak solutions

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References


C. Ya-Zhe and E. DiBenedetto, “Hölder estimates of solutions of singular parabolic equations with measurable coefficients,” Archive for rational mechanics and analysis, vol. 118, no. 3, pp. 257–271, 1992.

E. DiBenedetto, “On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients,” Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, vol. 13, no. 3, pp. 487–535, 1986.

V. Bögelein, “Global gradient bounds for the parabolic p-Laplacian system,” Proceedings of the London Mathematical Society, vol. 111, no. 3, pp. 633–680, 2015.

M. M. Porzio and V. Vespri, “Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations,” Journal of differential equations, vol. 103, no. 1, pp. 146–178, 1993.

E. DiBenedetto, J. M. Urbano, and V. Vespri, “Current issues on singular and degenerate evolution equations,” in Handbook of Differential Equations: Evolutionary Equations, vol. 1, Elsevier, 2002, pp. 169–286.

M. Misawa, “A Hölder estimate for nonlinear parabolic systems of p-Laplacian type,” Journal of Differential Equations, vol. 254, no. 2, pp. 847–878, 2013.

C. Karim and M. Misawa, “Hölder regularity for singular parabolic systems of $ p $-Laplacian type,” Advances in Differential equations, vol. 20, no. 7/8, pp. 741–772, 2015.

C. Karim and M. Misawa, “Gradient Hölder regularity for nonlinear parabolic systems of $ p $-Laplacian type,” Differential and Integral equations, vol. 29, no. 3/4, pp. 201–228, 2016.

M. Misawa, “Local Hölder regularity of gradients for evolutional p-Laplacian systems,” Annali di Matematica Pura ed Applicata, vol. 181, no. 4, pp. 389–405, 2002.

C. Karim and M. Misawa, “Existence of global weak solutions for Cauchy-Dirichlet problem for evolutional p-Laplacian systems,” in AIP Conference Proceedings, 2017, vol. 1913, no. 1, p. 020001.

C. Karim, “Local Boundedness of weak solutions for singular parabolic systems of p-Laplacian type,” The Australian Journal of Mathematical Analysis and Applications, vol. 15, no. 2–8, pp. 1–5, 2018.

E. DiBenedetto, Degenerate parabolic equations. Springer Science & Business Media, 2012.

P. R. Akbar, C. Karim, and R. B. EW, “The Existence of Global Weak Solutions for Singular Parabolic System of p-Laplacian Type,” in Journal of Physics: Conference Series, 2020, vol. 1562, no. 1, p. 012009.

C. Karim, “Local Hölder regularity of weak solutions for degenerate parabolic type,” in AIP Conference Proceedings, 2019, vol. 2192, no. 1, p. 050003.

J. M. Urbano, “The method of intrinsic scaling,” Lecture Notes in Mathematics, vol. 1930, 1930.

B. Dacorogna, Introduction to the Calculus of Variations. World Scientific Publishing Company, 2014.




DOI: https://doi.org/10.18860/ca.v7i1.13105

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