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Quasilinear parabolic problems in the Lebsgue-Sobolev space with variable exponent and $L^1$ data | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 10، دوره 15، شماره 10، دی 2024، صفحه 117-130 اصل مقاله (496.63 K) | ||
| نوع مقاله: Review articles | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2023.30528.4423 | ||
| نویسندگان | ||
| Souilah Fairouz1، 2؛ Maouni Messaoud* 2؛ Kamel Slimani* 2 | ||
| 1University 20th August 1955, Skikda, Algeria | ||
| 2Laboratory of Applied Mathematics and History and Didactics of Maths "LAMAHIS", Algeria | ||
| تاریخ دریافت: 11 اردیبهشت 1402، تاریخ بازنگری: 25 مرداد 1402، تاریخ پذیرش: 27 مرداد 1402 | ||
| چکیده | ||
| In this work, we study the existence of an initial boundary problem of a quasilinear parabolic problem with variable exponent and $ L ^{1} $-data of the type \begin{equation*} \left\{ \begin{array}{ll} (b(u))_{t}-\text{div}(\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u)+\lambda \left\vert u\right\vert ^{p(x)-2}u=f(x,t,u) \text{ } & \text{in}\hspace{0.5cm}Q=\Omega \times ]0,T[, \\ u=0 & \text{on}\hspace{0.5cm}\Sigma =\partial \Omega \times ]0,T[, \\ b(u)(t=0)=b(u_{0}) & \text{in}\hspace{0.5cm}\Omega , \end{array} \right. \end{equation*} where $ \lambda>0$ and $ T $ is positive constant. The main contribution of our work is to prove the existence of a renormalized solution. The functional setting involves Lebesgue– Sobolev spaces with variable exponents. | ||
| کلیدواژه‌ها | ||
| Quasilinear parabolic problems؛ variable exponent؛ truncations؛ renormalized solutions؛ L1 data | ||
| مراجع | ||
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[1] A. Aberqi, J. Bennouna, and M. Elmassoudi, Existence and uniqueness of renormalized solution for nonlinear parabolic equations in Musielak Orlicz spaces, Bol. Soc. Paran. Mat. 40 (2022), 1–22. [2] Y. Akdim, J. Bennouna, M. Mekkour, and H. Redwane, Existence of a renormalised solutions for a class of nonlinear degenerated parabolic problems with L1 data, J. Part. Diff. Eq. 26 (2013), no. 1, 7698. [3] E. Azroula, H. Redwane, and M. Rhoudaf, Existence of solutions for nonlinear parabolic systems via weak convergence of truncations, Electron. J. Differ. Equ. 2010 (2010), no. 68, 1–18. [4] M. Badr Benboubker, H. Chrayteh, M. EL Moumni, and H. Hjiaj, Entropy and renormalized solutions for nonlinear elliptic problem involving variable exponent and measure data, Acta Math. Sinica English Ser. 31 (2015), no. 1, 151–169. [5] M. Badr Benboubker, H. Hjiaj, I. Ibrango, and S. Ouaro, Existence of renormalized solutions for some quasilinear elliptic Neumann problems, Nonauton. Dyna. Syst. 8 (2021), no. 1, 180–206. [6] P. Benilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, and J. L. Vazquez, An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1995), no. 2, 241–273. [7] D. Blanchard, and F. Murat, Renormalised solutions of nonlinear parabolic problems with L1 data, Existence and uniqueness, Proc. Roy. Soc. Edin. Sect. A 127 (1997), 1137–1152. [8] D. Blanchard, F. Murat, and H. Redwane, Existence et unicite de la solution reormalisee d’un probl´eme parabolique assez general, C. R. Acad. Sci. Paris Ser. I 329 (1999), 575–580. [9] D. Blanchard, F. Murat, and H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differ. Equ. 177 (2001), 331–374. [10] L. Boccardo, J.I. Diaz, D. Giachetti, and F. Murat, Existence of a solution for a weaker form of a nonlinear elliptic equation, Recent advances in nonlinear elliptic and parabolic problems (Nancy, 1988)., volume 208 of Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1989, pp. 229–246. [11] L. Boccardo, A. Dall’Aglio, T. Gallouet, and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal. 147 (1997), 237–258. [12] T. M. Bendahmane, P. Wittbold, and A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1-data, J. Differ. Equ. 249 (2010), 1483–1515. [13] Y.M. Chen, S. Levine, and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383–1406. [14] R.-J. Di Perna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math. 130 (1989), 321–366. [15] B. El Hamdaoui, J. Bennouna, and A. Aberqi, Renormalized solutions for nonlinear parabolic systems in the Lebesgue Sobolev spaces with variable exponents, J. Math. Phys. Anal. Geom. 14 (2018), no. 1, 27–53. [16] A. Eljazouli and H. Redwane, Nonlinear elliptic system with variable exponents and singular coefficient and with diffuse measure data, Mediterr. J. Math. 18 (2021), article number: 107, 1–28. [17] S. Fairouz, M. Messaoud, and S. Kamel, Study of quasilinear parabolic problems with data L1, Proc. Int. Conf. Math. “An Istanbul Meeting for World Mathematicians”, Istanbul, Turkey. Los Press, 3-5 July 2019. [18] X.L. Fan and D. Zhao, On the spaces Lp(x)(U) and Wm;p(x)(U), J. Math. Anal. Appl. 263 (2001), 424-446. [19] R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary problems, Proc. Roy. Soc. Edin. Sect. A 89 (1981), 321–366. [20] J.-L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires, Dunod et Gauthier-Villars, 1969. [21] J.-L. Lions, Mathematical Topics in Fluid Mechanics, Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, New York, 1996. [22] F. Li, Z. Li, and L. Pi, Variable exponent functionals in image restoration, Appl. Math. Comput. 216 (2010), no. 3, 870–882. [23] A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura Appl. 177 (1999), no. 4, 143–172. [24] S. Ouaro and A. Ouedraogo, Nonlinear parabolic equation with variable exponent and L1-data, Electron. J. Differ. Equ. 2017 (2017), no. 32, 1–32. [25] H. Redwane, Existence of solution for a class of nonlinear parabolic systems, Electron. J. Qualit. Theory Differ. Equ. 2007 (2007), no. 24, 1–18. [26] J. Simon, Compact sets in Lp(0, T;B), Ann. Mat. Pura Appl. 146 (1987), 65–96. [27] F. Souilah, M. Maouni, and K. Slimani, The existence result of renormalized solution for nonlinear parabolic system with variable exponent and L1 data, Int. J. Anal. Appl. 18 (2020), no. 5, 748–773. [28] F. Souilah, M. Maouni, and K. Slimani, The existence of renormalized solution for quasilinear parabolic problem with variable exponents and measure data, J. Bol. Soc. Paran. Mat. 41 (2023), 1–27. [29] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics 1748, Springer, Berlin, 2000. [30] C. Zhang, Entropy solutions for nonlinear elliptic equations with variable exponents, Electron. J. Differ. Equ. 2014 (2014), no. 92, 1–14. [31] C. Zhang and S. Zhou, Renormalized and entropy solution for nonlinear parabolic equations with variable exponents and L1 data, J. Differ. Equ. 248 (2010) 1376-1400. [32] V.V. Zhikov, Averaging of functionals in the calculus of variations and elasticity theory, Izv. Ross. Akad. NaukSer. Mat. 50 (1986), no. 4, 675–710. [in Russian], English translation in Math. USSR-Izv. 29 (1987), no. 1, 33–66. | ||
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