Non-stability result of entropy solutions for nonlinear parabolic problems with singular measures

  • Mohammed AbdellaouiEmail author
  • Elhoussine Azroul


In this paper, we study the nonlinear parabolic equation given by
$$\begin{aligned} u_{t}-\text {div}(a(t,x,\nabla u))+|u|^{q-1}u=f+\lambda ,\quad \text { in }(0,T)\times \Omega , \end{aligned}$$
where \(1<p<N\), \(q>1\), \(f\in L^{1}(Q)\), \(\lambda \) is a measure concentrated on a set of zero parabolic r-capacity and \( u\mapsto -\text {div}(a(t,x,\nabla u))\) is a pseudo-monotone operator. We also consider the corresponding bilateral obstacle problem with measure data concentrated on a set of zero parabolic p-capacity whose model is
$$\begin{aligned} \langle u_{t}-\text {div}(a(t,x,\nabla u))-\lambda , v-u\rangle \ge 0, \end{aligned}$$
with \(u\in K=\lbrace w\in L^{p}(0,T;W^{1,p}_{0}(\Omega )): |w|\le 1\rbrace \) for every \(v\in K\). We define a notion of entropy solutions, we give convergence properties essential to our proofs and we establish a non-stability result.


Entropy solutions Non-stability Parabolic inequalities p-capacity Singular measures 

Mathematics subject classification

35K86 37K45 32U20 32D20 



Funding information is not applicable / No funding was received.

Compliance with ethical standards

Conflict of Interest

The authors would like to thank Pr. Francesco Petitta, Università di Roma, and anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. On behalf of all authors, the corresponding author states that there is no conflict of interest.

Informed consent

Informed consent was obtained from all individual participants included in the study.


  1. 1.
    Adams, R.: Sobolev spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Abdellaoui, M., Azroul, E.: Renormalized solutions to nonlinear parabolic problems with blowing up coefficients and general measure data, Ricerche di Matematica, pp 1–23 (2019)Google Scholar
  3. 3.
    Adams, D.R., Hedberg, L.I.: Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften, vol. 314. Springer, Berlin (1996)CrossRefGoogle Scholar
  4. 4.
    Aïssaoui, N.: Capacitary type estimates in strongly nonlinear potential theory and applications. Revista Matemática Complutense 14(2), 347–370 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aïssaoui, N.: Une théorie du potentiel dans les espaces d Orlicz, PhD Thesis, Fez, (1994)Google Scholar
  6. 6.
    Aïssaoui, N., Benkirane, A.: Capacités dans les espaces d’Orlicz. Ann. Sci. Math. Québec 18(1), 1–23 (1994)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brezis, H.: Nonlinear elliptic equations involving measures, contributions to nonlinear partial differential equations, Madrid, 1981, Pitman, Boston, MA, pp. 82–89 (1983)Google Scholar
  8. 8.
    Bénilan, P., Brezis, H.: Nonlinear problems related to the Thomas-Fermi equation. Dedicated to Philippe Bénilan. J. Evol. Equ. 3(4), 673–770 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bénilan, P., Brezis, H., Crandall, M.: A semilinear elliptic equation in \(L^{1}(\mathbb{R}^{N})\). Ann. Sc. Norm. Sup. Pisa Cl. Sci. 2, 523–555 (1975)zbMATHGoogle Scholar
  10. 10.
    Boccardo, L., Dall’Aglio, A., Gallouët, T., Orsina, L.: Nonlinear parabolic equations with measure data. J. Funct. Anal. 147, 237–258 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Boccardo, L., Gallouët, T., Orsina, L.: Existence and nonexistence of solutions for some nonlinear elliptic equations. J. Anal. Math. 73, 203–223 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Boccardo, L., Murat, F.: Increase of power leads to bilateral problems. In: Dal Maso, G., Dell’Antonio, G.F. (eds.) Composite media and homogenization theory, pp. 113–123. World Scientific, Singapore (1995)Google Scholar
  13. 13.
    Brezis, H., Nirenberg, L.: Removable singularities for nonlinear elliptic equations. Topol. Methods Nonlinear Anal. 9(2), 201–219 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Baras, P., Pierre, M.: Problèmes paraboliques semi-linéaires avec données mesures. Appl. Anal. 18, 111–149 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Blanchard, D., Porretta, A.: Nonlinear parabolic equations with natural growth terms and measure initial data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30, 583–622 (2001)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Baras, P., Pierre, M.: Singularités éliminables pour des équations semi-linéaires. Ann. Inst. Fourier (Grenoble) 34, 185–206 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Browder, F.E.: Existence theorems for nonlinear partial differential equations. Proc. Symp. Pure Math. 16, 1–60 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Brézis, H., Veron, L.: Removable singularities for some nonlinear elliptic equations. Arch. Rat. Mech. Anal. 75, 1–6 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.L.: An \(L^{1}-\)theory of existence and uniqueness of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22, 241–273 (1995)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Dall’Aglio, P., Leone, C.: Obstacles problems with measure data and linear operators. Potential Anal. 17, 45–64 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28, 741–808 (1999)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Dall’Aglio, A., Orsina, L.: On the limit of some nonlinear elliptic equations involving increasing powers. Asympt. Anal. 14, 49–71 (1997)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Droniou, J., Prignet, A.: Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data. No DEA 14(1–2), 181–205 (2007)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Droniou, J., Porretta, A., Prignet, A.: Parabolic capacity and soft measures for nonlinear equations. Potential Anal. 19(2), 99–161 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Fiorenza, A.: Orlicz capacities and applications to PDEs and Sobolev mappings. Progr. Nonlinear Differ. Equ. Their Appl. 63, 259–266 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fiorenza, A., Giannetti, F.: On Orlicz capacities and a nonexistence result for certain elliptic PDEs. Nonlinear Differ. Equ. Appl. 22, 1949–1958 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Fiorenza, A., Prignet, A.: Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data. ESAIM Control Optim. Calc. Var. 9, 317–341 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Fukushima, M., Sato, K., Taniguchi, S.: On the closable part of pre-Dirichlet forms and the fine supports of underlying measures. Osaka J. Math. 28, 517–535 (1991)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Klimsiak, T., Rozkosz, A.: On the structure of bounded smooth measures associated with a quasi-regular Dirichlet form. Bull. Polish Acad. Sci. Math. 65, 45–56 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Klimsiak, T., Rozkosz, A.: On the structure of diffuse measures for parabolic capacities, arXiv:1808.06422v2 [math.AP] 18 (Feb 2019)
  31. 31.
    Gallouët, T., Morel, J.M.: Resolution of a semilinear equation in \(L^1\). Proc. R. Soc. Edinb. 96, 275–288 (1984)CrossRefzbMATHGoogle Scholar
  32. 32.
    Leray, J., Lions, J.-L.: Quelques résultats de Višik sur les problèmes elliptiques semi-linéaires par les méthodes de Minty et Browder. Bull. Soc. Math. France 93, 97–107 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Orsina, L., Prignet, A.: Non-existence of solutions for some nonlinear elliptic equations involving measures. Proc. R. Soc. Edinb. Sect. A Math. 130(1), 167–187 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Pierre, M.: Parabolic capacity and Sobolev spaces. Siam J. Math. Anal. 14, 522–533 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Petitta, F., Porretta, A.: On the notion of renormalized solution to nonlinear parabolic equations with general measure data. J. Elliptic Parabol. Equ. 1, 201–214 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Petitta, F.: Renormalized solutions of nonlinear parabolic equations with general measure data. Ann. Mat. Pura ed Appl. 187(4), 563–604 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Petitta, F.: A non-existence result for nonlinear parabolic equations with singular measure data. Proc. R. Soc. Edin. Sect. A Math. 139, 381–392 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Petitta, F., Ponce, A.C., Porretta, A.: Approximation of diffuse measures for parabolic capacities. C. R. Acad. Sci. Paris, Ser. I 346, 161–166 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Petitta, F., Ponce, A.C., Porretta, A.: Diffuse measures and nonlinear parabolic equations. J. Evol. Equ. 11(4), 861–905 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Prignet, A.: Existence and uniqueness of entropy solutions of parabolic problems with \(L^{1}\) data. Nonliear Anal. TMA 28, 1943–1954 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Serrin, J., Zou, H.: Non-existence of positive solutions for the Lane-Emden system. Differ. Integral Equ, 9, 635–653 (1996)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Vázquez, J.L., Veron, L.: Removable singularities of some strongly nonlinear elliptic equations, Manuscripta Math. 33, 129–144 (1980/81)Google Scholar

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© Orthogonal Publisher and Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Laboratory LAMA, Faculty of Sciences Dhar El MahrazSidi Mohamed Ben Abdellah University of FezAtlas FezMorocco

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