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Harnack’s inequality for degenerate double phase parabolic equations under the non-logarithmic Zhikov’s condition

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Abstract

We prove Harnack-type inequalities for bounded non-negative solutions of the degenerate parabolic equations with (p, q) growth

$$ {u}_t-\operatorname{div}\left({\left|\nabla u\right|}^{p-2}\nabla u+a\left(x,t\right){\left|\nabla u\right|}^{q-2}\nabla u\right)=0,\kern0.5em a\left(x,t\right)\ge 0, $$

under the generalized non-logarithmic Zhikov’s conditions

$$ {\displaystyle \begin{array}{c}\begin{array}{cc}\left|a\left(x,t\right)-a\left(y,\tau \right)\right|\leqslant A\mu (r){r}^{q-p},& \left(x,t\right),\left(y,\tau \right)\in {Q}_{r,r}\left({x}_0,{t}_0\right),\end{array}\\ {}\begin{array}{ccc}\underset{r\to 0}{\lim}\mu (r){r}^{q-p}=0,& \underset{r\to 0}{\lim}\mu (r)=+\infty, & \underset{0}{\int }{\mu}^{-\beta }(r)\frac{dr}{r}\end{array}=+\infty, \end{array}} $$

with some β > 0.

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References

  1. Yu. A. Alkhutov and O. V. Krasheninnikova, “On the continuity of solutions of elliptic equations with a variable order of nonlinearity,” Tr. Mat. Inst. Steklova, 261, Differ. Uravn. i Din. Sist., 7–15 (2008); transl. in Proc. Steklov Inst. Math., 261, 1–10 (2008).

  2. Yu. A. Alkhutov and M. D. Surnachev, “Behavior at a boundary point of solutions of the Dirichlet problem for the p(x)-Laplacian,” Algebra i Analiz, 31(2), 88–117 (2019); transl. in St. Petersburg Math. J., 31(2), 251–271 (2020).

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

  4. S. Antontsev and V. Zhikov, “Higher integrability for parabolic equations of p(x, t)-Laplacian type,” Adv. Differential Equations, 10(9), 1053–1080 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Baroni and V. Bögelein, “Calderón-Zygmund estimates for parabolic p(x, t)-Laplacian systems,” Rev. Mat. Iberoam., 30(4), 1355–1386 (2014).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  9. V. Bögelein and F. Duzaar, “Hölder estimates for parabolic p(x, t)-Laplacian systems,” Math. Ann., 354(3), 907–938 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Bonafede and I. I. Skrypnik, “On Hölder continuity of solutions of doubly nonlinear parabolic equations with weight,” Ukr. Math. J., 51, 996–1012 (1999).

    Article  MATH  Google Scholar 

  11. K. O. Buryachenko and I. I. Skrypnik, “Local continuity and Harnack inequality for double-phase parabolic equations,” Potential Anal., 56, 137–164 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Colombo and G. Mingione, “Bounded minimisers of double phase variational integrals,” Arch. Rational Mech. Anal., 218(1), 219–273 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Colombo and G. Mingione, “Regularity for double phase variational problems,” Arch. Rational Mech. Anal., 215(2), 443–496 (2015).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Di Benedetto, U. Gianazza, and 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(9), 385–422 (2010).

    MathSciNet  MATH  Google Scholar 

  18. E. DiBenedetto, U. Gianazza, and V. Vespri, “A new approach to the expansion of positivity set of nonnegative solutions to certain singular parabolic partial differential equations,” Proc. Amer. Math. Soc., 138, 3521–3529 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  19. L. Diening, P. Harjulehto, P. Hästö, and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, Springer, Heidelberg, 2017.

  20. D. Mengyao, Z. Chao, and Z. Shulin, “Global boundedness and Hölder regularity of solutions to general p(x, t)-Laplace parabolic equations,” Math. Methods Appl. Sci., 43(9), 5809–5831 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  21. O. V. Hadzhy, I. I. Skrypnik, and M. V. Voitovych, “Interior continuity, continuity up to the boundary and Harnack’s inequality for double-phase elliptic equations with non-logarithmic growth,” Math. Nachrichten (in press).

  22. O. V. Hadzhy, M. O. Savchenko, I. I. Skrypnik, and M. V. Voitovych, On asymptotic behavior of solutions to non-uniformly elliptic equations with generalized Orlicz growth, arXiv:2208.05671v1 [math.AP] (2022).

  23. P. Harjulehto and P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces. Lecture Notes in Mathematics, vol. 2236, Springer, Cham (2019).

  24. P. Harjulehto, P. Hästö, V. Út and M. Lê, Nuortio, “Overview of differential equations with non-standard growth,” Nonlinear Anal., 72(12), 4551–4574 (2010).

  25. P. Harjulehto, P. Hästö, and M. Lee, “Hölder continuity of ω-minimizers of functionals with generalized Orlicz growth,” Ann. Scuola Norm. Sup. di Pisa, Cl. di Scienze, XXII(2), 549–582 (2021).

  26. S. Hwang and G. M. Lieberman, “Hölder continuity of bounded weak solutions to generalized parabolic p-Laplacian equations I: degenerate case,” Electron. J. Differential Equations, 2015(287), 1–32 (2015).

    MATH  Google Scholar 

  27. 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(288), 1–24 (2015).

  28. I. M. Kolodij, “On boundedness of generalized solutions of elliptic differential equations,” Vestnik Moskov. Gos. Univ., 1970(5), 44–52 (1970).

  29. I. M. Kolodij, “On boundedness of generalized solutions of parabolic differential equations,” Vestnik Moskov. Gos. Univ., 1971(5), 25–31 (1971).

  30. E. M. Landis, “Some questions in the qualitative theory of second-order elliptic equations (case of several independent variables),” Uspehi Mat. Nauk, 109(18)(1), 3–62 (1963).

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

  32. G. M. Lieberman, “The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations,” Comm. Partial Differential Equations, 16(2–3), 311–361 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  33. V. Liskevich and I. I. Skrypnik, “Isolated singularities of solutions to quasilinear elliptic equations,” Potential Analysis, 28(1), 1–16 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Marcellini, “Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions,” Arch. Rational Mech. Anal., 105(3), 267–284 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  35. P. Marcellini, “Regularity and existence of solutions of elliptic equations with p, q-growth conditions,” J. Differential Equations, 90(1), 1–30 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  36. V. G. Maz’ya, “Behavior near the boundary, of solutions of the Dirichlet problem for a second-order elliptic equation in divergent form,” Math. Notes of Ac. of Sciences of USSR, 2, 610–617 (1967).

    Google Scholar 

  37. M. O. Savchenko, I. I. Skrypnik, and Ye. A. Yevgenieva, “Continuity and Harnack inequalities for local minimizers of non uniformly elliptic functionals with generalized Orlicz growth under the non-logarithmic conditions,” Nonlinear analysis (in press).

  38. M. A. Shan, I. I. Skrypnik, and M. V. Voitovych, “Harnack’s inequality for quasilinear elliptic equations with generalized Orlicz growth,” Electr. J. Diff. Equ, 27, 1–16 (2021).

    MathSciNet  MATH  Google Scholar 

  39. M. A. Shan, “Removable isolated singularities for solutions of anisotropic porous medium equation,” Annali di Matematica Pure ed Applicata, 196, 1913–1926 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  40. A. E. Shishkov and Ye. A. Yevgenieva, “Localized blow-up regimes for quasilinear doubly degenerate parabolic equations,” Math. Notes, 106(4), 639–650 (2019).

  41. I. I. Skrypnik, “Harnack’s inequality for singular parabolic equations with generalized Orlicz growth under the non-logarithmic Zhikov’s condition,” J. Evol. Equ., 22, 45 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  42. I. I. Skrypnik and M. V. Voitovych, “B1 classes of De Giorgi-Ladyzhenskaya-Ural’tseva and their applications to elliptic and parabolic equations with generalized Orlicz growth conditions,” Nonlinear Anal., 202, 112–135 (2021).

  43. I. I. Skrypnik and M. V. Voitovych, “On the continuity of solutions of quasilinear parabolic equations with generalized Orlicz growth under non-logarithmic conditions,” Annali Mat. Pure Appl., 201, 1381–1416 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  44. I. I. Skrypnik and M. V. Voitovych, “B1 classes of De Giorgi-Ladyzhenskaya-Ural’tseva and their applications to elliptic and parabolic equations with generalized Orlicz growth conditions,” Nonlinear Anal., 202, 112–135 (2021).

  45. I. I. Skrypnik and Ye. A. Yevgenieva, Harnack inequality for solutions of the p(x)-Laplace equation under the precise non-logarithmic Zhikov’s conditions, arXiv.org/abs/2208.01970v1 [math.AP] (2022).

  46. M. D. Surnachev, “On Harnack’s inequality for p(x)-Laplacian,” Keldysh Institute Preprints, https://doi.org/10.20948/prepr-2018-69, 1–32 (2018).

  47. M. D. Surnachev, “On the weak Harnack inequality for the parabolic p(x)-Laplacian,” Asymptotic Anal., 1, 1–39 (2021).

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  51. F. Yao, “Hölder regularity of the gradient for the non-homogeneous parabolic p(x, t)-Laplacian equations,” Math. Methods Appl. Sci., 37(12), 1863–1872 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  52. F. Yao, “Hölder regularity for the general parabolic p(x, t)-Laplacian equations,” NoDEA Nonlinear Differential Equations Appl., 22(1), 105–119 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  53. Ye. A. Yevgenieva, “Propagation of singularities for large solutions of quasilinear parabolic equations,” J. Math. Phys. Anal. Geom., 15(1), 131–144 (2019).

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

    Article  MathSciNet  MATH  Google Scholar 

  55. V. V. Zhikov, “Questions of convergence, duality and averaging for functionals of the calculus of variations,” Izv. Akad. Nauk SSSR Ser. Mat., 47(5), 961–998 (1983).

    MathSciNet  MATH  Google Scholar 

  56. V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory,” Izv. Akad. Nauk SSSR Ser. Mat., 50(4), 675–710, 877 (1986).

  57. V. V. Zhikov, “On Lavrentiev’s phenomenon,” Russian J. Math. Phys., 3(2), 249–269 (1995).

    MathSciNet  MATH  Google Scholar 

  58. V. V. Zhikov, “On some variational problems,” Russian J. Math. Phys., 5(1), 105–116 (1998).

    MathSciNet  MATH  Google Scholar 

  59. V. V. Zhikov, “On the density of smooth functions in Sobolev-Orlicz spaces,” Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov (POMI), 310, Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts., 35[34], 67–81, 226 (2004); transl. in J. Math. Sci., 132(3), 285–294 (2006).

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

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk, Ukraine

    Mariia Savchenko, Igor Skrypnik & Yevgeniia Yevgenieva

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  1. Mariia Savchenko
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  2. Igor Skrypnik
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Correspondence to Mariia Savchenko.

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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 20, No. 1, pp. 124-155, January-March, 2023.

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Savchenko, M., Skrypnik, I. & Yevgenieva, Y. Harnack’s inequality for degenerate double phase parabolic equations under the non-logarithmic Zhikov’s condition. J Math Sci 273, 427–452 (2023). https://doi.org/10.1007/s10958-023-06508-9

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