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Journal of Evolution Equations

, Volume 19, Issue 2, pp 559–584 | Cite as

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

  • Kaimin Teng
  • Chao ZhangEmail author
  • Shulin Zhou
Article
  • 49 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  11. 11.
    D. Blanchard, H. Redwane, Renormalized solutions for a class of nonlinear evolution problems, J. Math. Pure Appl. 77 (1998) 117–151.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  14. 14.
    L. Boccardo, T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989) 149–169.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  17. 17.
    L. Caffarelli, Nonlocal equations, drifts and games, Nonlinear Partial Differ. Equ. Abel Symp. 7 (2012) 37–52.zbMATHGoogle Scholar
  18. 18.
    L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007) 1245–1260.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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 .MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  25. 25.
    J. Droniou, A. Porretta, A. Prignet, Parabolic capacity and soft measures for nonlinear equations, Potential Anal. 19 (2) (2003) 99–161.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  34. 34.
    C. Leone, A. Porretta, Entropy solutions for nonlinear elliptic equations in \(L^1\), Nonlinear Anal. 32 (3) (1998) 325–334.MathSciNetzbMATHGoogle Scholar
  35. 35.
    E. Lindgren, P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014) 795–826.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  42. 42.
    A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura Appl. 177 (1999) 143–172.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  44. 44.
    M. Sanchón, J. M. Urbano, Entropy solutions for the \(p(x)\)-Laplace equation, Trans. Amer. Math. Soc. 361 (2009) 6387–6405.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  48. 48.
    C. Zhang, S. Zhou, Renormalized solutions for a non-uniformly parabolic equation, Ann. Acad. Sci. Fenn. Math. 37 (2012) 175–189.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle 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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