Quasilinear parabolic equations with p(x)-Laplacian diffusion terms and nonlocal boundary conditions
DOI:
https://doi.org/10.24193/subbmath.2019.1.10Keywords:
Nonlocal boundary conditions, quasi-linear parabolic equations, generalized Lebesgue space and Sobolev spaces with variable exponents.Abstract
In this study, we prove the existence of local solution for a quasi linear generalized parabolic equation with nonlocal boundary conditions for an elliptic operator involving the variable-exponent nonlinearities, using Faedo-Galerkin arguments and compactness method.
Mathematics Subject Classification (2010): 35K20, 35K59, 35B45, 35D30.
References
Aboulaich, R., Meskine, D., Souissi, A., New diffusion models in image processing, Comput. Math. Appl., 56(2008), no. 4, 874-882.
Adams, R.A., Sobolev Spaces, Academic Press, 2003.
Antontsev, S.N., Shmarev, S.I., Elliptic Equations with Anisotropic Nonlinearity and Nonstandard Growth Conditions, Handbook of Differential Equations, Stationary Partial Differential Equations, vol. 3, 2006.
Antontsev, S.N., Shmarev, S.I., Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math., 234(2010), no. 9, 2633-2645.
Antontsev, S.N., Zhikov, V., Higher integrability for parabolic equations of p(x, t)- Laplacian type, Adv. Differential Equations, 10(2009), no. 9, 1053-1080.
Bouziani, A., Merazga, N., Benamira, S., Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions, Nonlinear Analysis, 69(2008), 1515- 1524.
Chen, B., Existence of solutions for quasilinear parabolic equations with nonlocal boundary conditions, Electronic Journal of Differential Equations, 2011(2011), no. 18, 1-9.
Chen, Y., Levine, S., Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66(2006), 1383-1406.
Diening, L., H¨asto, P., Harjulehto, P., Ru˜zicka, M., Lebesgue and Sobolev Spaces with Variable Exponents, in: Springer Lecture Notes, Springer-Verlag, vol. 2017, Berlin, 2011.
Diening, L., Ru˜zicka, M., Calderon Zygmund operators on generalized Lebesgue spaces Lp(x)(Ω) and problems related to fluid dynamics, J. Reine Angew. Math., 563(2003), 197-220.
Diening, L., Ru˜zicka, M., Calderon Zygmund operators on generalized Lebesgue spaces Lp(x)(Ω) and problems related to fluid dynamics, Preprint Mathematische Fakulta¨t, Albert-Ludwigs-Universit¨at Freiburg, 120(2002), 197-220.
Fan, X., Shen, J., Zhao, D., Sobolev embedding theorems for spaces Wk,p(x)(Ω), J. Math. Anal. Appl., 262(2001), 749-760.
Fu, Y., The existence of solutions for elliptic systems with nonuniform growth, Studia Math., 151(2002), 227-246.
Kov`acik, O., R´akosnik, J., On spaces Lp(x) and W 1,p(x)(Ω), 41(1991).
Lian, S., Gao, W., Cao, C., Yuan, H., Study of the solutions to a model porous medium equation with variable exponent of nonlinearity, J. Math. Anal. Appl., 342(2008), no. 1, 27-38.
Lions, J.L., Quelques m´ethodes de r´esolution des probl`emes aux limites non lin´eaires, Dunod, Paris, 1966.
Rahmoune, A., Semilinear Hyperbolic Boundary Value Problem Associated to the Nonlinear Generalized Viscoelastic Equations, Acta Mathematica Vietnamica, 43(2018), no. 2, 219-238.
Rahmoune, A., On the existence, uniqueness and stability of solutions for semilinear generalized elasticity equation with general damping, Acta Mathematica Sinica, English Series, 33(2017), no. 11, 1549-1564.
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