Acta mathematica scientia, Series B >
STRONGLY NONLINEAR VARIATIONAL PARABOLIC EQUATIONS WITH p(x)-GROWTH
Received date: 2015-04-24
Revised date: 2015-07-21
Online published: 2016-10-25
We study the Dirichlet problem associated to strongly nonlinear parabolic equations involving p(x) structure in W01,xLp(x)(Q). We prove the existence of weak solutions by applying Galerkin's approximation method.
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 . DOI: 10.1016/S0252-9602(16)30076-5
[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 Lp(x)(Ω) and Wm,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 Lp(x) and Wm,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∞(RN) in the generalized Sobolev spaces m,p(x)(RN)//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∞(RN) in the generalized Sobolev spaces m,p(x)(RN). Dokl Akad Nauk, 1999, 369(4):451-454
[20] Simon J. Compact set in the space Lp(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
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