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Renormalized and entropy solutions for the fractional p-Laplacian evolution equations

  • Kaimin Teng
  • Chao ZhangEmail author
  • Shulin Zhou
Article
  • 16 Downloads

Abstract

In this paper, we prove the existence and uniqueness of both nonnegative renormalized and entropy solutions for the fractional p-Laplacian evolution problems with nonnegative \(L^1\) data. In addition, we obtain the equivalence of renormalized and entropy solutions and establish a comparison result.

Keywords

Fractional p-Laplacian Renormalized solutions Entropy solutions Existence Uniqueness 

Mathematics Subject Classification

Primary 35D05 Secondary 35D10 46E35 

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Notes

Acknowledgements

The authors wish to thank the anonymous reviewer for valuable comments and suggestions to improve the expressions. K. Teng was supported by the NSFC (No. 11501403) and the Shanxi Province Science Foundation for Youths (No. 2013021001-3). C. Zhang was supported by the NSFC (No. 11671111) and Heilongjiang Province Postdoctoral Startup Foundation (LBH-Q16082). S. Zhou was supported by the NSFC (Nos. 11571020, 11671021).

References

  1. 1.
    B. Abdellaoui, A. Attar, R. Bentifour, On the fractional \(p\)-Laplacian equations with weight and general datum, Adv. Nonlinear Anal. (2016),  https://doi.org/10.1515/anona-2016-0072.
  2. 2.
    B. Abdellaoui, A. Attar, R. Bentifour, I. Peral, On fractional \(p\)-Laplacian parabolic problem with general data, Ann. Mat. Pura Appl. 197 (2) (2018) 329–356.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    N. Alibaud, B. Andreianov, M. Bendahmane, Renormalized solutions of the fractional Laplace equation, C. R. Acad. Sci. Paris, Ser. I 348 (2010) 759–762.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Alvino, L. Boccardo, V. Ferone, L. Orsina, G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura Appl. 182 (2003) 53–79.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    D. Applebaum, Lévy processes–from probability to finance quantum groups, Notices Amer. Math. Soc. 51 (11) (2004) 1336–1347.MathSciNetzbMATHGoogle Scholar
  6. 6.
    M. Bendahmane, P. Wittbold, A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and \(L^1\) data, J. Differential Equations 249 (6) (2010) 1483–1515.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J. L. Vazquez, An \(L^1\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1995) 241–273.MathSciNetzbMATHGoogle Scholar
  8. 8.
    D. Blanchard, F. Murat, Renormalised solutions of nonlinear parabolic problems with \(L^1\) data: Existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A 127 (6) (1997) 1137–1152.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D. Blanchard, F. Murat, H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differential Equations 177 (2) (2001) 331–374.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D. Blanchard, F. Petitta, H. Redwane, Renormalized solutions of nonlinear parabolic equations with diffuse measure data, Manuscripta Math. 141 (2013) 601–635.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    D. Blanchard, H. Redwane, Renormalized solutions for a class of nonlinear evolution problems, J. Math. Pure Appl. 77 (1998) 117–151.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    L. Boccardo, G. R. Cirmi, Existence and uniqueness of solution of unilateral problems with \(L^1\) data, J. Convex. Anal. 6 (1999) 195–206.MathSciNetzbMATHGoogle Scholar
  13. 13.
    L. Boccardo, A. Dall’Aglio, T. Gallouët, L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal. 147 (1997) 237–258.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    L. Boccardo, T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989) 149–169.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    L. Boccardo, T. Gallouët, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (5) (1996) 539–551.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    L. Boccardo, D. Giachetti, J. I. Diaz, F. Murat, Existence and regularity of renormalized solutions for some elliptic problems involving derivations of nonlinear terms, J. Differential Equations 106 (1993) 215–237.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    L. Caffarelli, Nonlocal equations, drifts and games, Nonlinear Partial Differ. Equ. Abel Symp. 7 (2012) 37–52.CrossRefzbMATHGoogle Scholar
  18. 18.
    L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007) 1245–1260.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    L. Caffarelli, E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 41 (2011) 203–240.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Y. Cai, S. Zhou, Existence and uniqueness of weak solutions for a non-uniformly parabolic equation, J. Funct. Anal. 257 (2009) 3021–3042.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. 12 (4) (1999) 741–808.MathSciNetzbMATHGoogle Scholar
  22. 22.
    A. Dall’Aglio, Approximated solutions of equations with \(L^1\) data. Application to the \(H\)-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl. 170 (4) (1996) 207–240.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012) 521–573 .MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    R. J. DiPerna, P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math. 130 (1989) 321–366.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    J. Droniou, A. Porretta, A. Prignet, Parabolic capacity and soft measures for nonlinear equations, Potential Anal. 19 (2) (2003) 99–161.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    J. Droniou, A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, NoDEA Nonlinear Differential Equations Appl. 14 (1-2) (2007) 181–205.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    V. G. Jakubowski, P. Wittbold, On a nonlinear elliptic–parabolic integro-differential equation with \(L^1\)-data, J. Differential Equations 197 (2) (2004) 427–445.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    K. H. Karlsen, F. Petitta, S. Ulusoy, A duality approach to the fractional Laplacian with measure data, Publ. Mat. 55 (1) (2011) 151–161.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    T. Klimsiak, A. Rozkosz, Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form, NoDEA Nonlinear Differential Equations Appl. 22 (6) (2015) 1911–1934.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    J. Korvenpää, T. Kuusi, E. Lindgren, Equivalence of solutions to fractional \(p\)-Laplace type equations, J. Math. Pures Appl. (2017),  https://doi.org/10.1016/j.matpur.2017.10.004.
  31. 31.
    T. Kuusi, G. Mingione, Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys. 337 (2015) 1317–1368.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    R. Landes, On the existence of weak solutions for quasilinear parabolic initial boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 89 (1981) 217–237.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    T. Leonori, I. Peral, A. Primo, F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst. 35 (12) (2015) 6031–6068.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    C. Leone, A. Porretta, Entropy solutions for nonlinear elliptic equations in \(L^1\), Nonlinear Anal. 32 (3) (1998) 325–334.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    E. Lindgren, P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014) 795–826.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible models, Oxford Univ. Press, Oxford, 1996.Google Scholar
  37. 37.
    J. M. Mazón, J. D. Rossi, J. Toledo, Fractional \(p\)-Laplacian evolution equations, J. Math. Pure Appl. 105 (2016) 810–844.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    R. Metzler, J. Klafter, The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004) 161–208.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    M. C. Palmeri, Entropy subsolutions and supersolutions for nonlinear elliptic equations in \(L^1\), Ricerche Mat. 53 (2004) 183–212.MathSciNetzbMATHGoogle Scholar
  40. 40.
    F. Petitta, Renormalized solutions of nonlinear parabolic equations with general measure data, Ann. Mat. Pura Appl. 187 (4) (2008) 563–604.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    F. Petitta, Some remarks on the duality method for integro-differential equations with measure data, Adv. Nonlinear Stud. 16 (1) (2016) 115–124.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura Appl. 177 (1999) 143–172.MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    A. Prignet, Existence and uniqueness of “entropy” solutions of parabolic problems with \(L^1\) data, Nonlinear Anal. 28 (12) (1997) 1943–1954.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    M. Sanchón, J. M. Urbano, Entropy solutions for the \(p(x)\)-Laplace equation, Trans. Amer. Math. Soc. 361 (2009) 6387–6405.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    K. Teng, Two nontrivial solutions for an elliptic problem involving some nonlocal integro-differential operators, Ann. Mat. Pura Appl. 194 (5) (2015) 1455–1468.MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    M. Xiang, B. Zhang, V. Radulescu, Existence of solutions for perturbed fractional \(p\)-Laplacian equations, J. Differential Equations 260 (2016) 1392–1413.MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    C. Zhang, S. Zhou, Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and \(L^1\) data, J. Differential Equations 248 (6) (2010) 1376–1400.MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    C. Zhang, S. Zhou, Renormalized solutions for a non-uniformly parabolic equation, Ann. Acad. Sci. Fenn. Math. 37 (2012) 175–189.MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    C. Zhang, S. Zhou, The well-posedness of renormalized solutions for a non-uniformly parabolic equation, Proc. Amer. Math. Soc. 145 (6) (2017) 2577–2589.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsTaiyuan University of TechnologyTaiyuanPeople’s Republic of China
  2. 2.Department of Mathematics, Institute for Advanced Study in MathematicsHarbin Institute of TechnologyHarbinPeople’s Republic of China
  3. 3.LMAM, School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China

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