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Intrinsic scaling method for doubly nonlinear parabolic equations and its application

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Published/Copyright: June 10, 2021

Abstract

In this article, we consider a fast diffusive type doubly nonlinear parabolic equation, called 𝑝-Sobolev type flows, and devise a new intrinsic scaling method to transform the prototype doubly nonlinear equation to the 𝑝-Sobolev type flows. As an application, we show the global existence and regularity for the 𝑝-Sobolev type flows with large data.

MSC 2010: 35B45; 35B65; 35D30; 35K61

Award Identifier / Grant number: 18K03375

Funding statement: The work by M. Misawa was partially supported by the Grant-in-Aid for Scientific Research (C) Grant number No. 18K03375 at Japan Society for the Promotion of Science.

Acknowledgements

We would like to record our sincere thanks to the referee for carefully reading this paper and giving some corrections.

  1. Communicated by: Juha Kinnunen

References

[1] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z. 183 (1983), no. 3, 311–341. 10.1007/BF01176474Search in Google Scholar

[2] T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. Search in Google Scholar

[3] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monogr. Math., Springer, Berlin, 1998. 10.1007/978-3-662-13006-3Search in Google Scholar

[4] P. Bénilan and M. G. Crandall, The continuous dependence on 𝜑 of solutions of u t - Δ φ ( u ) = 0 , Indiana Univ. Math. J. 30 (1981), no. 2, 161–177. 10.1512/iumj.1981.30.30014Search in Google Scholar

[5] T. Bhattacharya and L. Marazzi, On the viscosity solutions to Trudinger’s equation, NoDEA Nonlinear Differential Equations Appl. 22 (2015), no. 5, 1089–1114. 10.1007/s00030-015-0315-4Search in Google Scholar

[6] Y. M. Chen, M. C. Hong and N. Hungerbühler, Heat flow of 𝑝-harmonic maps with values into spheres, Math. Z. 215 (1994), no. 1, 25–35. 10.1007/BF02571698Search in Google Scholar

[7] E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J. 32 (1983), no. 1, 83–118. 10.1512/iumj.1983.32.32008Search in Google Scholar

[8] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer, New York, 1993. 10.1007/978-1-4612-0895-2Search in Google Scholar

[9] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack’s Inequality for Degenerate and Singular Parabolic Equations, Springer Monogr. Math., Springer, New York, 2012. 10.1007/978-1-4614-1584-8Search in Google Scholar

[10] L. C. Evans, Partial Differential Equations, Grad. Stud. in Math. 19, American Mathematical Society, Providence, 1998. Search in Google Scholar

[11] U. Gianazza and V. Vespri, Parabolic De Giorgi classes of order 𝑝 and the Harnack inequality, Calc. Var. Partial Differential Equations 26 (2006), no. 3, 379–399. 10.1007/s00526-006-0022-4Search in Google Scholar

[12] A. V. Ivanov, Uniform Hölder estimates for generalized solutions of quasilinear parabolic equations that admit double degeneration (in Russian), Algebra i Analiz 3 (1991), no. 2, 139–179; translation in St. Petersburg Math. J. 3 (1992), no. 2, 363–403. Search in Google Scholar

[13] A. V. Ivanov, Hölder estimates for a natural class of equations of fast diffusion type (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 229 (1995), no. 11, 29–62; translation in J. Math. Sci. (N. Y.) 89 (1998), no. 6, 1607–1630. Search in Google Scholar

[14] J. Kinnunen and T. Kuusi, Local behaviour of solutions to doubly nonlinear parabolic equations, Math. Ann. 337 (2007), no. 3, 705–728. 10.1007/s00208-006-0053-3Search in Google Scholar

[15] T. Kuusi, M. Misawa and K. Nakamura, Regularity estimates for the 𝑝-Sobolev flow, J. Geom. Anal. 30 (2020), no. 2, 1918–1964. 10.1007/s12220-019-00314-zSearch in Google Scholar

[16] T. Kuusi, M. Misawa and K. Nakamura, Global existence for the 𝑝-Sobolev flow, J. Differential Equations 279 (2021), 245–281. 10.1016/j.jde.2021.01.018Search in Google Scholar

[17] T. Kuusi, J. Siljander and J. M. Urbano, Local Hölder continuity for doubly nonlinear parabolic equations, Indiana Univ. Math. J. 61 (2012), no. 1, 399–430. 10.1512/iumj.2012.61.4513Search in Google Scholar

[18] E. Lindgren and P. Lindqvist, On a comparison principle for Trudinger’s equation, Adv. Calc. Var. (2020), 10.1515/acv-2019-0095. 10.1515/acv-2019-0095Search in Google Scholar

[19] M. Misawa, Local Hölder regularity of gradients for evolutional 𝑝-Laplacian systems, Ann. Mat. Pura Appl. (4) 181 (2002), no. 4, 389–405. 10.1007/s102310100044Search in Google Scholar

[20] M. Misawa, K. Nakamura and H. Sarkar, Extinction profile for a doubly nonlinear parabolic equation of fast diffusion type, preprint (2020). Search in Google Scholar

[21] K. Nakamura and M. Misawa, Existence of a weak solution to the 𝑝-Sobolev flow, Nonlinear Anal. 175 (2018), 157–172. 10.1016/j.na.2018.05.016Search in Google Scholar

[22] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations 103 (1993), no. 1, 146–178. 10.1006/jdeq.1993.1045Search in Google Scholar

[23] P. E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), no. 4, 387–409. 10.1016/0362-546X(83)90092-5Search in Google Scholar

[24] W. Schlag, Schauder and L p estimates for parabolic systems via Campanato spaces, Comm. Partial Differential Equations 21 (1996), no. 7–8, 1141–1175. 10.1080/03605309608821221Search in Google Scholar

[25] H. Schwetlick and M. Struwe, Convergence of the Yamabe flow for “large” energies, J. Reine Angew. Math. 562 (2003), 59–100. 10.1515/crll.2003.078Search in Google Scholar

[26] N. S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205–226. 10.1002/cpa.3160210302Search in Google Scholar

[27] V. Vespri, On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations, Manuscripta Math. 75 (1992), no. 1, 65–80. 10.1007/BF02567072Search in Google Scholar

[28] V. Vespri, Harnack type inequalities for solutions of certain doubly nonlinear parabolic equations, J. Math. Anal. Appl. 181 (1994), no. 1, 104–131. 10.1006/jmaa.1994.1008Search in Google Scholar

[29] V. Vespri and M. Vestberg, An extensive study of the regularity of solutions to doubly singular equations, Adv. Calc. Var. (2020), 10.1515/acv-2019-0102. 10.1515/acv-2019-0102Search in Google Scholar

[30] H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21–37. Search in Google Scholar

[31] R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom. 39 (1994), no. 1, 35–50. 10.4310/jdg/1214454674Search in Google Scholar

Received: 2020-11-16
Revised: 2021-03-12
Accepted: 2021-05-05
Published Online: 2021-06-10
Published in Print: 2023-04-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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