STRONGLY NONLINEAR VARIATIONAL PARABOLIC EQUATIONS WITH p(x)-GROWTH

doi:10.1016/S0252-9602(16)30076-5

数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (5): 1383-1404.doi: 10.1016/S0252-9602(16)30076-5

STRONGLY NONLINEAR VARIATIONAL PARABOLIC EQUATIONS WITH p(x)-GROWTH

Elhoussine AZROUL¹, Badr LAHMI¹, Ahmed YOUSSFI²

1. Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, Laboratory LAMA, P. O. Box 1796 Atlas Fès 30000, Morocco;
2. University Sidi Mohamed Ben Abdellah, National School of Applied Sciences, Laboratory of Engineering, Systems and Applications, P. O. Box 72-Principale, Fès 30000, Morocco

收稿日期:2015-04-24 修回日期:2015-07-21 出版日期:2016-10-25 发布日期:2016-10-25
通讯作者: Ahmed YOUSSFI,ahmed.youssfi@gmail.com E-mail:ahmed.youssfi@gmail.com
作者简介:Elhoussine AZROUL,azroulelhoussine@gmail.com;Badr LAHMI,lahmi.badr@gmail.com

STRONGLY NONLINEAR VARIATIONAL PARABOLIC EQUATIONS WITH p(x)-GROWTH

Elhoussine AZROUL¹, Badr LAHMI¹, Ahmed YOUSSFI²

1. Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, Laboratory LAMA, P. O. Box 1796 Atlas Fès 30000, Morocco;
2. University Sidi Mohamed Ben Abdellah, National School of Applied Sciences, Laboratory of Engineering, Systems and Applications, P. O. Box 72-Principale, Fès 30000, Morocco

Received:2015-04-24 Revised:2015-07-21 Online:2016-10-25 Published:2016-10-25
Contact: Ahmed YOUSSFI,ahmed.youssfi@gmail.com E-mail:ahmed.youssfi@gmail.com

摘要/Abstract

摘要：

We study the Dirichlet problem associated to strongly nonlinear parabolic equations involving p(x) structure in W₀^1,xL^p(x)(Q). We prove the existence of weak solutions by applying Galerkin's approximation method.

关键词: nonlinear parabolic equations, existence, variable exponents, weak solution

Abstract:

Key words: nonlinear parabolic equations, existence, variable exponents, weak solution

中图分类号:

35K55

Elhoussine AZROUL, Badr LAHMI, Ahmed YOUSSFI. STRONGLY NONLINEAR VARIATIONAL PARABOLIC EQUATIONS WITH p(x)-GROWTH[J]. 数学物理学报(英文版), 2016, 36(5): 1383-1404.

Elhoussine AZROUL, Badr LAHMI, Ahmed YOUSSFI. STRONGLY NONLINEAR VARIATIONAL PARABOLIC EQUATIONS WITH p(x)-GROWTH[J]. Acta mathematica scientia,Series B, 2016, 36(5): 1383-1404.

参考文献

[1] Adams R A, Fournier J J F. Sobolev Spaces. Pure and Appl Math 140. 2nd ed. Amsterdam:Elsevier/Academic Press, 2003
[2] Amann H. Ordinary Differential Equations:An Introduction to Nonlinear Analysis. Berlin, New York:De Gruyter, 1990
[3] Boccardo L, Murat F, Puel J P. Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann Mat Pura Appl, 1988, 152(4):183-196
[4] Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math, 2006, 66:1383-1406
[5] Diening L, Harjulehto P, Hästö P, R·u?i?cka M. Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, Vol 2017. Berlin:Springer, 2011
[6] Edmunds D E, Rákosn?k J. Sobolev embeddings with variable exponent. Stud Math, 2000, 143(3):267-293
[7] Elmahi A, Meskine D. Strongly nonlinear parabolic equations with natural growth terms in Orlicz spaces. Nonlinear Anal, 2005, 60:1-35
[8] Fu Y. The existence of solutions for elliptic systems with nonuniform growth. Studia Math, 2002, 151:227-246
[9] Fu Y, Pan N. Existence of solutions for nonlinear parabolic problem with p(x)-growth. J Math Anal Appl, 2010, 362:313-326
[10] Fan X, Zhao D. On the spaces L^p(x)(Ω) and W^m,p(x)(Ω). J Math Anal Appl, 2001, 263(2):424-446
[11] Harjulehto P, Hästö P, Koskenoja M, Varonen S. The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values. Potential Anal, 2006, 25:205-222
[12] Kovácik O, Rákosn?k J. On spaces L^p(x) and W^m,p(x). Czechoslovak Math J, 1991, 41:592-618
[13] 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
[14] Levine S, Chen Y, Stanich J. Image restoration via nonstandard diffusion. Technical Report:04-01, Dept of Mathematics and Computer Science, Duquesne University, 2004
[15] Musielak J. Orlicz Spaces and Modular Spaces. Lecture Notes in Math 1034. Berlin:Springer-Verlag, 1983
[16] Rajagopal K R, R·u?i?cka M. Mathematical modeling of electrorheological materials. Contin Mech Thermodyn, 2001, 13:59-78
[17] R·u?i?cka M. Electrorheological Fluids:Modeling and Mathematical Theory. Lecture Notes in Mathematics. Berlin:Springer, 2000
[18] Samko S G. Denseness of C₀^∞(R^N) in the generalized Sobolev spaces ^m,p(x)(R^N)//Direct and Inverse Problems of Mathematics Physics (Newark, DE, 1997), Int Soc Anal Appl Comput 5, Dordrecht:Kluwer Acad Publ, 2000:333-342
[19] Samko S G. Density of C₀^∞(R^N) in the generalized Sobolev spaces ^m,p(x)(R^N). Dokl Akad Nauk, 1999, 369(4):451-454
[20] Simon J. Compact set in the space L^p(0, T, B). Ann Mat Pura Appl, 1987, 146:65-96
[21] Zhang C, Zhou S. Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L¹-data. J Differ Equ, 2010, 248:1376-1400
[22] Zhikov V V. Density of smooth functions in Sobolev-Orlicz spaces. Zap Nauchn Sem S-Petersburg. Otdel Mat Inst Steklov (POMI), 2004, 310:67-81
[23] Zhikov V V. Averaging of functionals of the calculus of variations and elasticity theory.[in Russian], Izv Akad Nauk SSSR, Ser Mat, 1986, 50(4):675-710; English transl:Math USSR, Izv, 1987, 29(1):33-66
[24] Zhikov V V. On Lavrentiev's phenomen. Russian J Math Phy, 1995, 3(2):249-269

STRONGLY NONLINEAR VARIATIONAL PARABOLIC EQUATIONS WITH p(x)-GROWTH

STRONGLY NONLINEAR VARIATIONAL PARABOLIC EQUATIONS WITH p(x)-GROWTH

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