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Entropy solutions for a class of doubly nonlinear parabolic systems involving measure data in non-reflexive Orlicz spaces
Soluciones de entropía para una clase no lineal de sistemas parabólicos que involucran conjuntos de datos medibles en espacios de Orlicz no reflexivos
DOI:
https://doi.org/10.15446/recolma.v59n1.122857Palabras clave:
Parabolic systems, generalized growth, Orlicz-Sobolev spaces, Entropy solutions (en)Sistemas parabólicos, crecimiento generalizado, espacios de Orlicz-Sobolev, soluciones de entropía (es)
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This paper deals with an existence result of entropy solutions for a nonlinear parabolic systems of the form
∂ui/∂t - div (a(x, t, ui, ∇ui) + Φi(x, t, ui)) = fi(x, u1, u2) - div(Fi) in QT
ui = 0 on Γ
ui(t = 0) = ui,0 in Ω,
where the lower order term Φ satisfies a growth condition prescribed by the Nfunction M defining the framework spaces (see section 2.1) and the right hand side is a measure datum. The main term which contains the space derivatives and a non-coercive lower order term are considered in divergence form satisfying only the original Orlicz growths. We don’t assume any restriction neither on M nor on its complementary M. Therefor, we work in a nonreflexive Orlicz spaces.
Este artículo se centra en probar la existencia de soluciones entrópicas para sistemas no lineales parabólicos de la forma
∂ui/∂t - div (a(x, t, ui, ∇ui) + Φi(x, t, ui)) = fi(x, u1, u2) - div(Fi) in QT
ui = 0 on Γ
ui(t = 0) = ui,0 in Ω,
donde el término Φ satisface una condición de crecimiento dada por la N-función M definida en los espacios descritos en la sección 2.1 y el lado derecho de la ecuación corresponde a las condiciones medibles asociadas al problema. El término principal que contiene los términos con derivadas espaciales y los términos no coercitivos de menor orden, que aparecen forma divergente, satisfacen las tasas de crecimiento de Orlicz. No vamos a suponer ninguna restricción sobre la función M o la función complementaria M. Por lo tanto, trabajaremos en espacios de Orlicz no reflexivos.
Referencias
[1] R. Adams, Sobolev spaces, Academic Press Inc, New York, 1975.
[2] Y. Ahmida, I. Chlebicka, P. Gwiazda, and A. Youssfi, Gossez’s approximation theorems in Musielak-Orlicz-Sobolev spaces, J. Funct. Anal., 2018, https://doi.org/10.1016/j.jfa.2018.05.015.
[3] M. Ben Cheikh Ali and O. Guib´e, Nonlinear and non-coercive elliptic problems with integrable data, (English summary). Adv. Math. Sci. Appl. 16 (2006), no. 1, 275-297.
[4] E. Azroul, H. Redwane, and M. Rhoudaf, Existence of solutions for nonlinear parabolic systems via weak convergence of truncations, Elec. J. of Diff. Equ. 2010 (2010), no. 68, 1-18.
[5] A. Eden, B. Michaux, and J. Rakotoson, Doubly Nonlinear Parabolic-Type Equations as Dynamical systems, Journal of Dynamics and Differential Equations 3 (1991), 87-131.
[6] A. Elmahi and D. Meskine, Strongly nonlinear parabolic equations with natural growth terms and L1 data in Orlicz spaces, Portugaliae Mathematica. Nova 62 (2005), 143-183.
[7] A. Elmahi and D. Meskine, Strongly nonlinear parabolic equations with natural growth terms in Orlicz spaces, Nonlinear Anal. Theory Methods Appl. 60 (2005), 1-35.
[8] J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. soc. 190 (1974), 163-205.
[9] J. P. Gossez, Some approximation properties in Orlicz-Sobolev spaces, Stud. Math. 74 (1982), 17-24.
[10] J. P. Gossez and V. Mustonen, Variationnal inequalities in Orlicz-Sobolev spaces, Nonlinear Anal. 11 (1987), 317-492.
[11] A. EL Hachimi and H. EL Ouardi, Existence and regularity of a global attractor for doubly nonlinear parabolic equtions, Electron. J. Diff. Eqns. 45 (2002), 1-15.
[12] A. EL Hachimi and H. EL Ouardi, Attractors for a Class of Doubly Nonlinear Parabolic Systems, Electron. J. Diff. Eqns. 1 (2006), 1-15.
[13] M. Krasnosel’skii and Y. Rutikii, Convex functions and Orlicz spaces, Groningen, Nordhooff, 1969.
[14] M. Mabdaoui, H. Moussa, and M. Rhoudaf, Entropy solutions for a nonlinear parabolic problems with lower order term in Orlicz spaces, Anal.Math.Phys., 2016, DOI 10.1007/s13324-016-0129-5.
[15] M. Marion, Attractors for reaction-diffusion equation: existence of their dimension, Applicable Analysis 25 (1987), 101-147.
[16] H. Moussa and M. Rhoudaf, Existence of renormalized solution of nonlinear elliptic problems with lower order term in Orlicz spaces, Ricerche mat., 2017, DOI 10.1007/s11587-017-0322-3 249.
[17] R. Di Nardo, F. Feo, and O. Guibé, Existence result for nonlinear parabolic equations with lower order terms, Anal. Appl. (Singap.) 9 (2011), no. 2, 161-186.
[18] A. Porretta, Existence results for strongly nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura Appl. 177 (1999), no. IV, 143-172.
[19] H. Redwane, Existence of a solution for a class of nonlinear parabolic systems, Elect. J. Qual. Th. Diff. Equ. 24 (2007), 18pp.
[20] H. Redwane, Existence Results For a Class of Nonlinear Parabolic Equations in Orlicz Spaces, Electronic Journal of Qualitative Theory of Differential Equations 2 (2010), 1-19.
[21] J. Simon, Compact sets in Lp(0, T;B), Ann. Mat. Pura Appl. 146 (1987), 65-96.
