[1] Adams R A, Fournier J F. Sobolev Spaces.Amsterdam: Academic Press, 2003
[2] Afraites L, Atlas A, Karami F, Meskine D. Some class of parabolic systems applied to image processing. Discrete Contin Dyn Syst Ser B, 2016, 21(6): 1671-1687
[3] Ahmida Y.On Some Functional Analysis Results in Musielak Spaces and Applications to PDEs. Fez: University of Sidi Mohamed Ben Abdellah, 2019
[4] Ahmida Y, Chlebicka I, Gwiazda P, Youssfi A. Gossez's approximation theorems in Musielak-Orlicz-Sobolev spaces. J Funct Anal, 2018, 275(9): 2538-2571
[5] Ahmida Y, Fiorenza A, Youssfi A. $H = W$ Musielak spaces framework. Atti Accad Naz Lincei Rend Lincei Mat Appl, 2020, 31(2): 447-464
[6] Ahmida Y, Youssfi A.Some approximation results in time and space dependent Musielak spaces. arXiv: 2404.10158
[7] Ahmida Y, Youssfi A. Poincaré-type inequalities in Musielak spaces. Ann Acad Sci Fenn Math, 2019, 44: 1041-1054
[8] Ahmida Y, Youssfi A. Variational nonlinear elliptic equations in nonreflexive Musielak spaces. J Math Anal Appl, 2020, 491(2): Art 124387
[9] Antontsev S, Shmarev S.Parabolic equations with anisotropic nonstandard growth conditions//Figueiredo I N, Rodrigues J F, Santos L. Free Boundary Problems. Basel: Birkhäuser, 2007
[10] Antontsev S, Shmarev S. Anisotropic parabolic equations with variable nonlinearity. Publ Mat, 2009, 53(2): 355-399
[11] Azroul E, Lahmi B, Youssfi A. Strongly nonlinear variational parabolic equations with $p(x)$-growth. Acta Math Sci, 2016, 36B(5): 1383-1404
[12] Bulíček M, Gwiazda P, Skrzeczkowski J. Parabolic equations in Musielak-Orlicz spaces with discon-tinuous in time $N$-function. J Differential Equations, 2021, 290: 17-56
[13] Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math, 2006, 66(4): 1383-1406
[14] Chlebicka I, Gwiazda P, Zatorska-Goldstein A. Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak-Orlicz spaces in the class of renormalized solutions. J Differential Equations, 2018, 265(11): 5716-5766
[15] Chlebicka I, Gwiazda P, Zatorska-Goldstein A. Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon. Ann Inst H Poincaré Anal Non Linéaire, 2019, 36(5): 1431-1465
[16] Chlebicka I, Gwiazda P, Zatorska-Goldstein A. Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon. J Differential Equations, 2019, 267(2): 1129-1166
[17] Diening L, Nägele P, R$\mathring{\rm u}$žička M. Monotone operator theory for unsteady problems in variable exponent spaces. Complex Var Elliptic Equ, 2012, 57(11): 1209-1231
[18] Donaldson T. Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems. J Differential Equations, 1974, 16: 201-256
[19] Elmahi A, Meskine D. Parabolic equations in Orlicz spaces. J London Math Soc, 2005, 72(2): 410-428
[20] Elmahi A, Meskine D. Strongly nonlinear parabolic equations with natural growth terms in Orlicz spaces. Nonlinear Anal, 2005, 60(1): 1-35
[21] Gwiazda P, Świerczewska-Gwiazda A, Wróblewska A. Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids. Math Methods Appl Sci, 2010, 33(2): 125-137
[22] Gwiazda P, Wittbold P, Wróblewska-Kamińska A, Zimmermann A. Renormalized solutions to nonlinear parabolic problems in generalized Musielak-Orlicz spaces. Nonlinear Anal, 2015, 129: 1-36
[23] Hale J K. Ordinary Differential Equations.Malabar: Robert E Krieger Publishing Company, 1980
[24] Harjulehto P, Hästö P. Orlicz Spaces and Generalized Orlicz Spaces. Heidelberg: Springer, 2019
[25] Hudzik H. On problem of density of C$_0^\infty$($\Omega$) in generalized Orlicz-Sobolev space W$^k_M$($\Omega$) for every open set $\Omega\subset \mathbb{R}^n$. Comment Math Prace Mat, 1977, 20(1): 65-78
[26] Hudzik H. Density of C$_0^\infty$($\mathbb{R}^n$) in generalized Orlicz-Sobolev space W$^k_M$($\mathbb{R}^n$). Funct Approx Comment Math, 1979, 7: 15-21
[27] Kamińska A. Some convexity properties of Musielak-Orlicz spaces of Bochner type//Proceedings of the 13th Winter School on Abstract Analysis. Circolo Matematico di Palermo, 1985
[28] Kamińska A, Kubiak D. The Daugavet property in the Musielak-Orlicz spaces. J Math Anal Appl, 2015, 427(2): 873-898
[29] Landes R. On the existence of weak solutions for quasilinear parabolic initial-boundary value problems. Proc Roy Soc Edinburgh Sect A, 1981, 89(3/4): 217-237
[30] Landes R, Mustonen V. A strongly nonlinear parabolic initial-boundary value problem. Ark Mat, 1987, 25(1): 29-40
[31] Le Dret H.Nonlinear Elliptic Partial Differential Equations. Cham: Springer, 2018
[32] Li Y, Zhang C.Renormalized and entropy solutions to the general nonlinear parabolic equations in Musielak-Orlicz spaces. arXiv: 2411.01828
[33] Lions J L.Quelques Méthodes de Résolution des Probl$\grave{\rm e}$mes aux Limites Non-Linéaires. Paris: Dunod, 1969
[34] Maeda F Y. Poincaré type inequalities for variable exponents. JIPAM J Inequal Pure Appl Math, 2008, 9(3): Art 68
[35] Maeda F Y, Mizuta Y, Ohno T, Shimomura T. Approximate identities and Young type inequalities in Musielak-Orlicz spaces. Czechoslovak Math J, 2013, 63: 933-948
[36] Maeda F Y, Mizuta Y, Ohno T, Shimomura T. Boundedness of maximal operators and Sobolev's inequality on Musielak-Orlicz-Morrey spaces. Bull Sci Math, 2013, 137(1): 76-96
[37] Menovschikov A, Molchanova A, Scarpa L. An extended variational theory for nonlinear evolution equations via modular spaces. SIAM J Math Anal, 2021, 53(4): 4865-4907
[38] Musielak J.Orlicz Spaces and Modular Spaces. Berlin: Springer-Verlag, 1983
[39] Nägele P. Monotone Operators in Spaces with Variable Exponents. Freiburg: Universität Freiburg, 2009
[40] Rajagopal K, R$\mathring{\rm u}$žička M. On the modeling of electrorheological materials. Mech Res Commun, 1996, 23: 401-407
[41] Świerczewska-Gwiazda A. Nonlinear parabolic problems in Musielak-Orlicz spaces. Nonlinear Anal, 2014, 98: 48-65
[42] Youssfi A, Ahmida Y. Some approximation results in Musielak-Orlicz spaces. Czechoslovak Math J, 2020, 70: 453-471
[43] Zeidler E.Nonlinear Functional Analysis and its Applications. New York: Springer-Verlag, 1990
[44] Zhang C, Zhou S. Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and $L_1$ data. J Differential Equations, 2017, 248(6): 1376-1400
