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Harnack’s inequality for singular parabolic equations with generalized Orlicz growth under the non-logarithmic Zhikov’s condition

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Abstract

For a general class of divergence type quasi-linear singular parabolic equations with generalized Orlicz growth, we prove the intrinsic Harnack inequality for positive solutions. This class of singular equations includes new cases of equations with (p, q) nonlinearity and non-logarithmic growth.

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References

  1. Yu. A. Alkhutov, O. V. Krasheninnikova, On the continuity of solutions of elliptic equations with a variable order of nonlinearity, (Russian) Tr. Mat. Inst. Steklova 261 (2008), Differ. Uravn. i Din. Sist., 7–15; translation in Proc. Steklov Inst. Math. 261 (2008), no. 1–10.

  2. Yu. A. Alkhutov, M. D. Surnachev, Behavior at a boundary point of solutions of the Dirichlet problem for the \(p(x)\)-Laplacian, (Russian) Algebra i Analiz 31 (2019), no. 2, 88–117; translation in St. Petersburg Math. J. 31 (2020), no. 2, 251–271.

  3. Yu. A. Alkhutov, V. V. Zhikov, Hölder continuity of solutions of parabolic equations with variable nonlinearity exponent, Translation of Tr. Semin. im. I. G. Petrovskogo No. 28 (2011), Part I, 8–74; J. Math. Sci. (N.Y.) 179 (2011), no. 3, 347–389.

  4. S. Antontsev, V. Zhikov, Higher integrability for parabolic equations of \(p(x,t)\)-Laplacian type, Adv. Differential Equations 10 (2005), no. 9, 1053–1080.

  5. P. Baroni, V. Bögelein, Calderón-Zygmund estimates for parabolic \(p(x, t)\)-Laplacian systems, Rev. Mat. Iberoam. 30 (2014), no. 4, 1355–1386.

  6. P. Baroni, M. Colombo, G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. 121 (2015), 206–222.

  7. P. Baroni, M. Colombo, G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J. 27 (2016), 347–379.

  8. P. Baroni, M. Colombo, G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations 57 (2018), Paper No. 62, 48 pp.

  9. V. Bögelein, F. Duzaar, Hölder estimates for parabolic \(p(x, t)\)-Laplacian systems, Math. Ann. 354 (2012), no. 3, 907–938.

  10. K. O. Buryachenko, I. I. Skrypnik, Local Continuity and Harnack Inequality for Double-Phase Parabolic Equations, Potential Anal. (2020), https://doi.org/10.1007/s11118-020-09879-9

  11. Y. Z. Chen, E. Di Benedetto, Hölder estimates of solutions of singular parabolic equations with measurable coefficients, Arch. Rational Mech. Anal. 118 (1992), 257–271.

  12. M. Colombo, G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Rational Mech. Anal. 218 (2015), no. 1, 219–273.

  13. M. Colombo, G. Mingione, Regularity for double phase variational problems, Arch. Rational Mech. Anal. 215 (2015), no. 2, 443–496.

  14. M. Colombo, G. Mingione, Calderon-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal. 270 (2016), 1416–1478.

  15. E. Di Benedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.

  16. E. Di Benedetto, U. Gianazza, V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math. 200 (2008), 181–209.

  17. E. Di Benedetto, U. Gianazza, V. Vespri, Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5 (2010), no. 9, 385–422.

  18. L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, in: Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011, x+509 pp.

  19. Ding Mengyao, Zhang Chao, Zhou Shulin, Global boundedness and Hölder regularity of solutions to general \(p(x,t)\)-Laplace parabolic equations, Math. Methods Appl. Sci. 43 (2020), no. 9, 5809–5831.

  20. O. V. Hadzhy, I. I. Skrypnik, M. V. Voitovych, Interior continuity, continuity up to the boundary and Harnack’s inequality for double-phase elliptic equations with non-logarithmic growth, ar**v:201210960v1 [math.AP].

  21. P. Harjulehto, P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces, in: Lecture Notes in Mathematics, vol. 2236, Springer, Cham, 2019, p. X+169 https://doi.org/10.1007/978-3-030-15100-3

  22. P. Harjulehto, P. Hästö, Út V. Lê, M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), no. 12, 4551–4574.

  23. S. Hwang, G. M. Lieberman, Hölder continuity of bounded weak solutions to generalized parabolic \(p\)-Laplacian equations I: degenerate case, Electron. J. Differential Equations, 2015 (2015), no. 287, 1–32.

  24. S. Hwang, G. M. Lieberman, Hölder continuity of bounded weak solutions to generalized parabolic \(p\)-Laplacian equations II: singular case, Electron. J. Differential Equations, 2015 (2015), no. 288, 1–24.

  25. I. M. Kolodij, On boundedness of generalized solutions of elliptic differential equations, Vestnik Moskov. Gos. Univ. 1970 (1970), no. 5, 44–52.

  26. I. M. Kolodij, On boundedness of generalized solutions of parabolic differential equations, Vestnik Moskov. Gos. Univ. 1971 (1971), no. 5, 25–31.

  27. E. M. Landis, Some questions in the qualitative theory of second-order elliptic equations (case of several independent variables), Uspehi Mat. Nauk, 109 (1963), 18, no. 1, 3–62 (in Russian).

  28. E. M. Landis, Second Order Equations of Elliptic and Parabolic Type, in: Translations of Mathematical Monographs, vol. 171, American Math. Soc., Providence, RI, 1998.

  29. G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), no. 2-3, 311–361.

  30. P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions, Arch. Rational Mech. Anal. 105 (1989), no. 3, 267–284.

  31. P. Marcellini, Regularity and existence of solutions of elliptic equations with \(p,q\)-growth conditions, J. Differential Equations 90 (1991), no. 1, 1–30.

  32. M. A. Shan, I. I. Skrypnik, M. V. Voitovych, Harnack’s inequality for quasilinear elliptic equations with generalized Orlicz growth, Electron. J. Differential Equations 2021 (2021), no. 27, 1–16.

  33. I. I. Skrypnik, M. V. Voitovych, \(\cal{B}_{1}\) classes of De Giorgi-Ladyzhenskaya-Ural’tseva and their applications to elliptic and parabolic equations with generalized Orlicz growth conditions, Nonlinear Anal. 202 (2021) 1121-1135.

  34. I. I. Skrypnik, M. V. Voitovych, On the continuity of solutions of quasilinear parabolic equations with generalized Orlicz growth under non-logarithmic conditions, Ann.Mat.Pura Appl., https://doi.org/10.1007/510231-021-01161-y.

  35. M. D. Surnachev, On Harnack’s inequality for \(p(x)\)-Laplacian (Russian), Keldysh Institute Preprints 10.20948/prepr-2018-69, 69 (2018), 1–32.

  36. M. D. Surnachev, On the weak Harnack inequality for the parabolic \(p(x)\)- Laplacian, Asymptotic Analysis, https://doi.org/10.3233/ASY-211746.

  37. M. V. Voitovych, Pointwise estimates of solutions to 2m-order quasilinear elliptic equations with m-(p,q) growth via Wolff potentials, Nonlinear Anal. 181 (2019), 147–179.

  38. Y. Wang, Intrinsic Harnack inequalities for parabolic equations with variable exponents, Nonlinear Anal. 83 (2013) 12–30.

  39. P. Winkert, R. Zacher, Global a priori bounds for weak solutions to quasilinear parabolic equations with nonstandard growth, Nonlinear Anal. 145 (2016), 1–23.

    Article  MathSciNet  Google Scholar 

  40. M. Xu, Y. Chen, Hölder continuity of weak solutions for parabolic equations with nonstandard growth conditions, Acta Math. Sin. (Engl. Ser.) 22(3) (2006) 793–806.

  41. F. Yao, Hölder regularity of the gradient for the non-homogeneous parabolic \(p(x, t)\)-Laplacian equations, Math. Methods Appl. Sci., 37 (2014), no. 12, 1863–1872.

    Article  MathSciNet  Google Scholar 

  42. F. Yao, Hölder regularity for the general parabolic \(p(x, t)\)-Laplacian equations, NoDEA Nonlinear Differential Equations Appl. 22 (2015), no. 1, 105–119.

    Article  MathSciNet  Google Scholar 

  43. C. Zhang, S. Zhou, X. Xue, Global gradient estimates for the parabolic \(p(x, t)\)-Laplacian equation, Nonlinear Anal. 105 (2014) 86–101.

    Article  MathSciNet  Google Scholar 

  44. V. V. Zhikov, Questions of convergence, duality and averaging for functionals of the calculus of variations, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 5, 961–998.

  45. V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710, 877.

  46. V. V. Zhikov, On Lavrentiev’s phenomenon, Russian J. Math. Phys. 3 (1995), no. 2, 249–269.

  47. V. V. Zhikov, On some variational problems, Russian J. Math. Phys. 5 (1997), no. 1, 105–116 (1998).

  48. V. V. Zhikov, On the density of smooth functions in Sobolev-Orlicz spaces, (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34], 67–81, 226; translation in J. Math. Sci. (N.Y.) 132 (2006), no. 3, 285–294.

  49. V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994.

  50. V. V. Zhikov, S. E. Pastukhova, On the improved integrability of the gradient of solutions of elliptic equations with a variable nonlinearity exponent, (Russian) Mat. Sb. 199 (2008), no. 12, 19–52; translation in Sb. Math. 199 (2008), no. 11–12, 1751–1782.

  51. V. V. Zhikov, S. E. Pastukhova, On the property of higher integrability for parabolic systems of variable order of nonlinearity, (Russian) Mat. Zametki 87 (2010), no. 2, 179–200; translation in Math. Notes 87 (2010), no. 1–2, 169–188.

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Acknowledgements

This work was supported by Ministry of Education and Science of Ukraine (Project Number 0121U109525) and by the Volkswagen Foundation project ”From Modeling and Analysis to Approximation”.

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Gen. Batiouk Str. 19, Sloviansk, 84116, Ukraine

    Igor I. Skrypnik

  2. Vasyl’ Stus Donetsk National University, 600-richcha Str. 21, Vinnytsia, 21021, Ukraine

    Igor I. Skrypnik

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Correspondence to Igor I. Skrypnik.

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In memory of DiBenedetto.

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Skrypnik, I.I. Harnack’s inequality for singular parabolic equations with generalized Orlicz growth under the non-logarithmic Zhikov’s condition. J. Evol. Equ. 22, 45 (2022). https://doi.org/10.1007/s00028-022-00794-7

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