Nonlinear parabolic systems in Musielack-Orlicz space
Resumo
In this paper, we discuss the solvability of the nonlinear parabolic systems associated to the nonlinear parabolic equation: for $i=1,2$
\begin{equation*}
\mathcal{(S)} \left\{\begin{array}{lll}
\displaystyle\frac{\partial u_{i}}{\partial t} -div(a(x,t,u_{i},\nabla u_{i}))+ g_{i}(x,t,u_{i},\nabla u_i)) =f_{i}(x,u_{1},u_{2})\, \quad & \mbox{in}\quad & Q_{T},\\
\displaystyle u_{i}(x,t)=0 &\mbox{on}& \partial\Omega \times(0,T),\\
\displaystyle u_{i}(x,(t=0))=u_{i,0}(x) & \mbox{in} & \Omega,
\end{array}%
\right.
\end{equation*}
with the source $f$ is merely integrable. The operator $\displaystyle A(u)= div \Big (a(x,t,u_{i},\nabla u_{i})\Big)$
is a generalized Leray-Lions operator defined on the inhomogeneous Musielak-Orlicz spaces (the vector field $\displaystyle a(x,t,u_i,\nabla u_i)$ have a growth prescribed by a generalized N-function). The non linearity $g_{i}$ is a Carath\'{e}odory function satisfy the some condition.
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Referências
R. A. Adams; Sobolev spaces; New York (NY): Academic Press; 1975.
R. Adams; On the Orlicz-Sobolev imbeding theorem; J. Func. Analysis, 24, (1977), 241-257. https://doi.org/10.1016/0022-1236(77)90055-6
L. Aharouch and J. Bennouna; Existence and uniqueness of solutions of unilateral problems in Orlicz spaces. Nonlinear Anal. 72, (2010), 3553-3565. https://doi.org/10.1016/j.na.2010.01.005
A. Aberqi, J. Bennouna and H. Redwane; Existence result for a class of doubly nonlinear parabolic systems; Appl.Math(Warsaw), (2014), 1-11. https://doi.org/10.4064/am41-2-7
A. Aberqi, J. Bennouna and M. Hammoumi; Existence Result for Nonlinear Degenerated ParabolicSystems; Nonlinear Dynamics and Systems Theory, 17 (3) (2017) 217-229.
A. Benkirane and M. Sidi El Vally; Variational inequalities in Musielak-Orlicz-Sobolevspaces; Bull. Belg. Math. Soc. Simon Stevin, pp., (2014), 787-811. https://doi.org/10.36045/bbms/1420071854
T. Donaldson and N. S. Trudinger; Orlicz-Sobolev spaces and imbeding theorem; J.Functioal Analysis, 8(1971), 52-75. https://doi.org/10.1016/0022-1236(71)90018-8
R.-J. Diperna and P.-L. Lions; On the Cauchy Problem for the Boltzmann Equations: Global existence and weak stability; Ann. of Math. 130 (1989), 285-366. https://doi.org/10.2307/1971423
A. El hachimi and E. Elouardi; Existence and regularity of global attractor for doubly nonlinear parabolic equations; EJDE 45 (2002), 1-15.
A. El hachimi and E. Elouardi; Attractor for a class of doubly nonlinear parabolic systems; EJDE, 1 (2006), 1-15. https://doi.org/10.14232/ejqtde.2006.1.1
M. Elmassoudi, A. Aberqi and J. Bennouna, Existence of Entropy Solutions in Musielak-Orlicz Spaces Via a Sequence of Penalized Equations, Bol. Soc. Paran. Mat.https://doi.org/10.5269/bspm.v38i6.37269
M. El Moumni; Nonlinear elliptic equations without sign condition and L1 -data in Musielak-Orlicz-Sobolev spaces, Acta. Appl. Math. 159:(2019) 95-117. https://doi.org/10.1007/s10440-018-0186-x
M. El Moumni; Renormalized solutions for strongly nonlinear elliptic problems with lower order terms and measure data in Orlicz-Sobolev spaces, Iran. J. Math. Sci. Inform. Vol 14, No 1, (2019)pp 95-119.
J. P. Gossez; Some approximation properties in Orlicz-Sobolev spaces; Studia Mathematica, T. LXXIV.(1982). https://doi.org/10.4064/sm-74-1-17-24
O . Kovacik and J. Rakosnik; On spaces Lp(x) (Ω) and W1,p(x) (Ω); CzechoslovakMath. J. 41(116) (1991), 592-618. https://doi.org/10.21136/CMJ.1991.102493
Ait Khellou M. , Benkirane A., Elliptic inequalities with L1 data in Musielak-Orlicz spaces, Monatsh Math., 183-133, (2017). https://doi.org/10.1007/s00605-016-1010-1
J. Musielak; Modular spaces and Orlicz spaces; Lecture Notes in Math. (1983). https://doi.org/10.1007/BFb0072210
K. Rajagopal and M. Ruziecka; On the modeling of electrorheological materials; Mech. Res. Comm. 23 (1996), 401-407. https://doi.org/10.1016/0093-6413(96)00038-9
H. Redwane; existance of solution for a classe of nonlinear parabolic systems; Electron. J . Qual. Theory Differ. Equ., 24(2007), 18pp. https://doi.org/10.14232/ejqtde.2007.1.24
J. Rakotoson, A. Eden and B. Michaux; Doubly nonlinear parabolic-type equations as dynamical systems; Journal of dynamics and differential equations . 3 (1991). https://doi.org/10.1007/BF01049490
J. Serrin; Pathological solution of elliptic differential equations; Ann. Sc. Norm. Super. Pisa Cl. Sci. 18(1964), 385-387.
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