On some nonlinear parabolic equations with variable exponents and measure data

Hamdaoui, Bouchra El; Bennouna, Jaouad

doi:10.2478/mjpaa-2020-0008

Abstract

We prove the existence of renormalized solutions to a class of nonlinear evolution equations, supplemented with initial and Dirichlet condition in the framework of generalized Sobolev spaces. The data are assumed merely integrable.

References

[1] A. Aberqi, J. Bennouna, M. Mekkour and H. Redwane, Nonlinear parabolic inequalities with lower order terms, Applicable Analysis(2016), DOI: 10.1080/00036811.2016.120518610.1080/00036811.2016.1205186
Search in Google Scholar Back to article
[2] A. Almeida, P. Hasto, Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258 no. 5, 162-165 (2010).
Search in Google Scholar Back to article
[3] A. Almeida, P. Hasto, Interpolation in variable exponent spaces, to appear in Rev. Mat. Complut.
Search in Google Scholar Back to article
[4] E. Azroul, H. Hjiaj and B. Lahmi, Existence of entropy solutions for some strongly nonlinear p(x) -parabolic problems with L1-data, An. Univ. Craiova Ser. Mat. Inform. 42(2) (2015) 273-299.
Search in Google Scholar Back to article
[5] M. Bendahmane, P. Wittbold, Renormalized solutions for nonlinear elliptic equations with variable exponents and L1data, Nonlinear Anal. 70 (2) (2009) 567–583.10.1016/j.na.2007.12.027
Search in Google Scholar Back to article
[6] Ph. Bénilan, L. Boccardo, Th. Gallouët, R. Gariepy, M. Pierre, J.L. Vazquez, An L1 theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 22, 241-273, (1995)
Search in Google Scholar Back to article
[7] D. Blanchard, F. Murat, Renormalized solutions of nonlinear parabolic with L1 data: existence and uniqueness. Proc. Roy. Soc. Edinburgh Sect, A 127, pp. 1137-1152, 1997.
Search in Google Scholar Back to article
[8] L. Boccardo, D. Giachetti, J.-I. Diaz, F. Murat, Existence and regularity of renormalized solutions of some elliptic problems involving derivatives of nonlinear terms. Journal of differential equations 106, pp. 215-237, 1993.10.1006/jdeq.1993.1106
Search in Google Scholar Back to article
[9] M. Bendahmane, P. Wittbold, A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1-data. J. Differential Equation. 249, pp. 483-515 (2010).
Search in Google Scholar Back to article
[10] D. Blanchard, F. Murat, H. Redwane, Existence and uniqueness of renormalized solution for fairly general class of non linear parabolic problems, J.Differential Equations.No. 177, 331-374, 2001.10.1006/jdeq.2000.4013
Search in Google Scholar Back to article
[11] D. Blanchard, H. Redwane, Renormalized solutions for class of nonlinear evolution problems, J. Math. Pure. 77, pp. 117-151, 1998.10.1016/S0021-7824(98)80067-6
Search in Google Scholar Back to article
[12] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006) 1383–1406.10.1137/050624522
Search in Google Scholar Back to article
[13] G. Dal Maso, F. Murat, L. Orsina, and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuala Norm. Sup. Pisa Cl.Sci. (4), 28, (1999), 741808.
Search in Google Scholar Back to article
[14] T. Del Vecchio, M. R. Posteraro, An existence result for nonlinear and noncoercive problems. Nonlinear Anal., 31(1-2); 191–206, (1998).10.1016/S0362-546X(96)00304-5
Search in Google Scholar Back to article
[15] R.-J. Diperna, P.-L. Lions:On the Cauchy Problem for the Boltzmann Equations: Global existence and weak stability, ann. of Math. 130, pp. 285-366, 1989.10.2307/1971423
Search in Google Scholar Back to article
[16] R. Di Nardo, F. Fo, O. Guib, Existence result for nonlinear parabolic equations with lower order terms. Anal. Appl. 9(2), 161186 (2011).
Search in Google Scholar Back to article
[17] R. Di Nardo, A. Perrotta, Uniquness results for nonlinear elliptic problems ith two lowers order terms. Bull. Sci. Math. 137, 107128 (2013).
Search in Google Scholar Back to article
[18] B. Elhamdaoui, J. Bennouna, A. Aberqi, Renormalized Solutions for Nonlinear Parabolic Systems in the Lebesgue Sobolev Spaces with Variable Exponents Journal of Mathematical Physics, Analysis, Geometry 2018, Vol. 14, No. 1, pp. 1-2610.15407/mag14.01.027
Search in Google Scholar Back to article
[19] X.L. Fan, D. Zhao, On the spaces Lp(.)and Wm,p(.), J. Math. Anal. Appl. 263 (2001)424-446.10.1006/jmaa.2000.7617
Search in Google Scholar Back to article
[20] X.L Fan, D. Zhao. the generalised Orlicz-Sobolev space Wk,p(x)(Ω). J. Gansu Educ, College 12(1)1998, 1-6.
Search in Google Scholar Back to article
[21] R. Landes: On the existence of weak solutions for quasilinear parabolic initial-boundary problems, Proc. Roy. Soc. Edinburgh Sect. A89, pp. 321-366, 1981.10.1017/S0308210500020242
Search in Google Scholar Back to article
[22] J.-L. Lions, Quelques méthodes de resolution des problémes aux limites non linéaires, Dunod et Gauthier-Villars, Paris, 1969.
Search in Google Scholar Back to article
[23] P. Harjulehto. Variable exponent Sobolev space with zero boundary values. Math. Bohem.132(2007)125-136.
Search in Google Scholar Back to article
[24] O. Kovacik, J. Rakonsik, on space Lp(x)(Ω) and W1,p(x)(Ω). czechoslovak Math J. 41(116)1991592-618.
Search in Google Scholar Back to article
[25] L. Nirenberg, On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa, 13, 116-162, (1959).
Search in Google Scholar Back to article
[26] A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of trauncations, Ann. Mat. Pura ed Applicata, 177, pp. 143-172, (1999).10.1007/BF02505907
Search in Google Scholar Back to article
[27] H. Redwane, Existence of Solution for a class of a parabolic equation with three unbounded nonlinearities, Adv. Dyn. Syt. A2PL. 2, pp. 241-264, 2007.
Search in Google Scholar Back to article
[28] J.-M. Rakotoson, Resolution of the critical cases for the problems with L1 data, Asymptotic Analysis when right hand side is in L1(Ω). 6, pp. 285-293, 1993.10.3233/ASY-1993-6305
Search in Google Scholar Back to article
[29] H. Redwane, Solution renormalisé de problémes paraboliques et elliptiques non linéaires. Ph.D., Rouen 1997
Search in Google Scholar Back to article
[30] O. Kováčik and j.Rákosnĭk On spaces Lp(x)and W1,p(x), Czechoslovak Math, J. 41 (116) (1991) 592–618.10.21136/CMJ.1991.102493
Search in Google Scholar Back to article
[31] J. Simon, Compact set in the space Lp(0, T, B), Ann. Mat. Pura, Appl. 146, pp. 65-96, (1987).10.1007/BF01762360
Search in Google Scholar Back to article
[32] P. Wittbold, A. Zimmermann, Existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and L1data, Nonlinear Anal. 72 (2010) 2990–300810.1016/j.na.2009.11.041
Search in Google Scholar Back to article
[33] V. V. Zhikov, On Lavrentiev’s phenomenon, Russian Journal of Mathematical Physics, 3, 249-269, (1994).
Search in Google Scholar Back to article
[34] V.V. Zhikov, On the density of smooth functions in Sobolev-Orlicz spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004) 67–81.
Search in Google Scholar Back to article
[35] L. Boccardo, D. Giachetti, J. I. Diaz and F. Murat, Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. J. Differential Equations 106, no. 2, 215237, (1993).10.1006/jdeq.1993.1106
Search in Google Scholar Back to article
[36] A. DallAglio, Approximated solutions of equations with L1 data. Application to the Hconvergence of quasi-linear parabolic equations. Ann. Mat. Pura Appl. (4) 170, 207240, (1996).10.1007/BF01758989
Search in Google Scholar Back to article
[37] F. Murat, Solutions renormalizadas de edp elipticas no lineales. Publ. Laboratoire dAnalyse Numrique, Univ. Paris 6, R 93023, (1993).
Search in Google Scholar Back to article

On some nonlinear parabolic equations with variable exponents and measure data

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