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Keywords:
$p(x)$-Laplacian; Weak solution; Variable exponents.
Summary:
In this work, we study the existence and uniqueness of weak solutions of fourth-order degenerate parabolic equation with variable exponent using the difference and variation methods.
References:
[1] Andreianov, B., Bendahmane, M., Ouaro, S.: Structural stability for variable exponent elliptic problems, I: The $p(x)$-Laplacian kind problems. Nonlinear Anal., 73, 2010, 2-24, DOI 10.1016/j.na.2010.02.039 | MR 2645827 | Zbl 1191.35126
[2] Ansini, L., Giacomelli, L.: Shear-thinning liquid films: macroscopic and asymptotic behavior by quasi-self-similar solutions. Nonlinearity, 15, 2002, 2147-2164, DOI 10.1088/0951-7715/15/6/318 | MR 1938485
[3] Ansini, L., Giacomelli, L.: Doubly nonlinear thin-film equations in one space dimension. Arch. Ration. Mech. Anal., 173, 2004, 89-131, DOI 10.1007/s00205-004-0313-x | MR 2073506 | Zbl 1064.76012
[4] Antontsev, S.N., Shmarev, S.I.: A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal., 60, 2005, 515-545, DOI 10.1016/j.na.2004.09.026 | MR 2103951 | Zbl 1066.35045
[5] Antontsev, S., Shmarev, S.: Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions. Handbook of Differential Equations: Stationary Partial Differential Equations, 3, 2006, 1-100, Zbl 1192.35047
[6] Antontsev, S., Shmarev, S.: Parabolic equations with anisotropic nonstandard growth conditions. Internat. Ser. Numer. Math., 154, 2006, 33-44, DOI 10.1007/978-3-7643-7719-9_4 | MR 2305342
[7] Antontsev, S., Shmarev, S.: Blow-up of solutions to parabolic equations with nonstandard growth conditions. J. Comput. Appl. Math., 234, 2010, 2633-2645, DOI 10.1016/j.cam.2010.01.026 | MR 2652114 | Zbl 1196.35057
[8] Antontsev, S., Shmarev, S.: Vanishing solutions of anisotropic parabolic equations with variable nonlinearity. J. Math. Anal. Appl., 361, 2010, 371-391, DOI 10.1016/j.jmaa.2009.07.019 | MR 2568702 | Zbl 1183.35177
[9] Bertsch, M., Giacomelli, L., Lorenzo, G., Karali, G.: Thin-film equations with Partial wetting energy: Existence of weak solutions. Physica D, 209, 2005, 17-27, DOI 10.1016/j.physd.2005.06.012 | MR 2167440 | Zbl 1079.76011
[10] Bhuvaneswari, V., Shangerganesh, L., Balachandran, K.: Weak solutions for $p$-Laplacian equation. Adv. Nonlinear Anal., 1, 2012, 319-334, MR 3037124 | Zbl 1277.35117
[11] Bowen, M., Hulshof, J., King, J. R.: Anomalous exponents and dipole solutions for the thin film equation. SIAM J. Appl. Math., 62, 2001, 149-179, DOI 10.1137/S0036139900366936 | MR 1857540 | Zbl 1001.35074
[12] Cahn, J. W., Hilliard, J. E.: Free energy of nonuniform system I. interfacial free energy. J. Chem. Phys., 28, 1958, 258-367, DOI 10.1063/1.1744102
[13] Calderon, C. P., Kwembe, T. A.: Dispersal models. Rev. Union Mat. Argentina, 37, 1991, 212-229, MR 1266684 | Zbl 0795.92029
[14] Chang, K.: Critical Point Theory and Its Applications. 1986, Shangai Sci. Tech. Press, Shangai, MR 0865982 | Zbl 0698.58002
[15] Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math., 66, 2006, 1383-1406, DOI 10.1137/050624522 | MR 2246061 | Zbl 1102.49010
[16] Diening, L., Harjulehto, P., Hasto, P., Ruzicka, M.: Lebesgue and Sobolev Spaces With Variable Exponents. 2011, Springer-Verlag, Heidelberg, MR 2790542 | Zbl 1222.46002
[17] Evans, L. C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. 1990, American Mathematical Society, Providence, RI, MR 1034481 | Zbl 0698.35004
[18] Gao, W., Guo, Z.: Existence and localization of weak solutions of nonlinear parabolic equations with variable exponent of nonlinearity. Ann. Mat. Pura Appl., 191, 2012, 551-562, MR 2958349 | Zbl 1272.35135
[19] Guo, Z., Liu, Q., Sun, J., Wu, B.: Reaction-diffusion systems with $p(x)$-growth for image denoising. Nonlinear Anal. RWA, 12, 2011, 2904-2918, MR 2813233 | Zbl 1219.35340
[20] King, J. R.: Two generalization of the thin film equation. Math. Comput. Modeling, 34, 2001, 737-756, DOI 10.1016/S0895-7177(01)00095-4 | MR 1858796
[21] Lions, J.: Quelques Methodes de Resolution des Problems aux Limites Non lineaire. 1969, Dunod Editeur Gauthier Villars, Paris,
[22] Liu, C.: Some properties of solutions for the generalized thin film equation in one space dimension. Boletin de la Asociacion Matematica venezolana, 12, 2005, 43-52, MR 2192402 | Zbl 1099.35115
[23] Liu, C., Yin, J., Gao, H.: On the generalized thin film equation. Chin. Ann. Math., 25, 2004, 347-358, DOI 10.1142/S0252959904000329 | MR 2086127
[24] Ruzicka, M.: Electrorheological Fluids: Modeling and Mathematical Theory. 1748, 2000, Springer-Verlag, Berlin, MR 1810360 | Zbl 0968.76531
[25] Xu, M., Zhou, S.: Existence and uniqueness of weak solutions for a generalized thin film equation. Nonlinear Anal., 60, 2005, 755-774, DOI 10.1016/j.na.2004.01.013 | MR 2109157 | Zbl 1073.35102
[26] Xu, M., Zhou, S.: Stability and regularity of weak solutions for a generalized thin film equation. J. Math. Anal. Appl., 337, 2008, 49-60, DOI 10.1016/j.jmaa.2007.03.075 | MR 2356053 | Zbl 1124.35025
[27] Zang, A., Fu, Y.: Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces. Nonlinear Anal., 69, 2008, 3629-3636, DOI 10.1016/j.na.2007.10.001 | MR 2450565 | Zbl 1153.26312
[28] Zhang, C., Zhou, S.: A fourth-order degenerate parabolic equation with variable exponent. J. Part. Diff. Eq., 2009, 1-16, MR 2589555 | Zbl 1212.35259
[29] Zhang, C., Zhou, S.: Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and $L^1$ data. J. Differential Equations, 248, 2010, 1376-1400, DOI 10.1016/j.jde.2009.11.024 | MR 2593046 | Zbl 1195.35097
[30] Zhou, S.: A priori $L^{\infty}$-estimate and existence of weak solutions for some nonlinear parabolic equations. Nonlinear Anal., 42, 2000, 887-904, DOI 10.1016/S0362-546X(99)00135-2 | MR 1776929
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