On some nonlinear parabolic equations with variable exponents and measure data

[1] A. Aberqi, J. Bennouna, M. Mekkour and H. Redwane, Nonlinear parabolic inequalities with lower order terms, Applicable Analysis(2016), DOI: 10.1080/00036811.2016.120518610.1080/00036811.2016.1205186Search in Google Scholar

[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

[3] A. Almeida, P. Hasto, Interpolation in variable exponent spaces, to appear in Rev. Mat. Complut.Search in Google Scholar

[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

[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.2007.12.027Search in Google Scholar

[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

[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

[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.1993.1106Search in Google Scholar

[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

[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.2000.4013Search in Google Scholar

[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)80067-6Search in Google Scholar

[12] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006) 1383–1406.10.1137/050624522Search in Google Scholar

[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

[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)00304-5Search in Google Scholar

[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, 1989.10.2307/1971423Search in Google Scholar

[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

[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

[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/mag14.01.027Search in Google Scholar

[19] X.L. Fan, D. Zhao, On the spaces L^p(.)and W^m,p(.), J. Math. Anal. Appl. 263 (2001)424-446.10.1006/jmaa.2000.7617Search in Google Scholar

[20] X.L Fan, D. Zhao. the generalised Orlicz-Sobolev space W^k,p(x)(Ω). J. Gansu Educ, College 12(1)1998, 1-6.Search in Google Scholar

[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/S0308210500020242Search in Google Scholar

[22] J.-L. Lions, Quelques méthodes de resolution des problémes aux limites non linéaires, Dunod et Gauthier-Villars, Paris, 1969.Search in Google Scholar

[23] P. Harjulehto. Variable exponent Sobolev space with zero boundary values. Math. Bohem.132(2007)125-136.Search in Google Scholar

[24] O. Kovacik, J. Rakonsik, on space L^p(x)(Ω) and W^1,p(x)(Ω). czechoslovak Math J. 41(116)1991592-618.Search in Google Scholar

[25] L. Nirenberg, On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa, 13, 116-162, (1959).Search in Google Scholar

[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/BF02505907Search in Google Scholar

[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

[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-1993-6305Search in Google Scholar

[29] H. Redwane, Solution renormalisé de problémes paraboliques et elliptiques non linéaires. Ph.D., Rouen 1997Search in Google Scholar

[30] O. Kováčik and j.Rákosnĭk On spaces L^p(x)and W^1,p(x), Czechoslovak Math, J. 41 (116) (1991) 592–618.10.21136/CMJ.1991.102493Search in Google Scholar

[31] J. Simon, Compact set in the space L^p(0, T, B), Ann. Mat. Pura, Appl. 146, pp. 65-96, (1987).10.1007/BF01762360Search in Google Scholar

[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.2009.11.041Search in Google Scholar

[33] V. V. Zhikov, On Lavrentiev’s phenomenon, Russian Journal of Mathematical Physics, 3, 249-269, (1994).Search in Google Scholar

[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

[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.1993.1106Search in Google Scholar

[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/BF01758989Search in Google Scholar

[37] F. Murat, Solutions renormalizadas de edp elipticas no lineales. Publ. Laboratoire dAnalyse Numrique, Univ. Paris 6, R 93023, (1993).Search in Google Scholar