Well-Posedness and Numerical Study for Solutions of a Parabolic Equation with Variable-Exponent Nonlinearities
Jamal H. Al-Smail
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261, Saudi Arabia kfupm.edu.sa
Search for more papers by this authorCorresponding Author
Salim A. Messaoudi
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261, Saudi Arabia kfupm.edu.sa
Search for more papers by this authorAla A. Talahmeh
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261, Saudi Arabia kfupm.edu.sa
Search for more papers by this authorJamal H. Al-Smail
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261, Saudi Arabia kfupm.edu.sa
Search for more papers by this authorCorresponding Author
Salim A. Messaoudi
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261, Saudi Arabia kfupm.edu.sa
Search for more papers by this authorAla A. Talahmeh
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261, Saudi Arabia kfupm.edu.sa
Search for more papers by this authorAbstract
We consider the following nonlinear parabolic equation: ut − div(|∇u|p(x)−2∇u) = f(x, t), where and the exponent of nonlinearity p(·) are given functions. By using a nonlinear operator theory, we prove the existence and uniqueness of weak solutions under suitable assumptions. We also give a two-dimensional numerical example to illustrate the decay of solutions.
References
- 1 Alaoui M. K., Messaoudi S. A., and Khenous H. B., A blow-up result for nonlinear generalized heat equation, Computers & Mathematics with Applications. An International Journal. (2014) 68, no. 12, part A, 1723–1732, https://doi.org/10.1016/j.camwa.2014.10.018, MR3283362.
- 2 Pinasco J. P., Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. (2009) 71, no. 3-4, 1094–1099, https://doi.org/10.1016/j.na.2008.11.030, MR2527528.
- 3
Shangerganesh L.,
Gurusamy A., and
Balachandran K., Weak solutions for nonlinear parabolic equations with variable exponents, Communications in Mathematics. (2017) 25, no. 1, 55–70, MR3667076, 2-s2.0-85022089992, https://doi.org/10.1515/cm-2017-0006.
10.1515/cm-2017-0006 Google Scholar
- 4 Acerbi E. and Mingione G., Regularity results for a class of functionals with non-standard growth, Archive for Rational Mechanics and Analysis. (2001) 156, no. 2, 121–140, MR1814973, https://doi.org/10.1007/s002050100117, Zbl0984.49020, 2-s2.0-0035579913.
- 5 Acerbi E. and Mingione G., Regularity results for stationary electro-rheological fluids, Archive for Rational Mechanics and Analysis. (2002) 164, no. 3, 213–259, MR1930392, https://doi.org/10.1007/s00205-002-0208-7, Zbl1038.76058, 2-s2.0-0036768849.
- 6 Chen Y., Levine S., and Rao M., Variable exponent, linear growth functionals in image restoration, SIAM Journal on Applied Mathematics. (2006) 66, no. 4, 1383–1406, MR2246061, https://doi.org/10.1137/050624522, Zbl1102.49010, 2-s2.0-33747163537.
- 7 Diening L., Ettwein F., and Ruzicka M., C1,α-regularity for electrorheological fluids in two dimensions, Nonlinear Differential Equations and Applications NoDEA. (2007) 14, no. 1-2, 207–217, https://doi.org/10.1007/s00030-007-5026-z, MR2346460.
- 8 Ettwein F. and Ruzicka M., Existence of local strong solutions for motions of electrorheological fluids in three dimensions, Computers & Mathematics with Applications. An International Journal. (2007) 53, no. 3-4, 595–604, https://doi.org/10.1016/j.camwa.2006.02.032, MR2323712.
- 9 Guo Z., Liu Q., Sun J., and Wu B., Reaction-diffusion systems with p(x)-growth for image denoising, Nonlinear Analysis: Real World Applications. (2011) 12, no. 5, 2904–2918, https://doi.org/10.1016/j.nonrwa.2011.04.015, MR2813233.
- 10 Diening L., Harjulehto P., Hästö P., and Růžička M., Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics. (2011) 2017, 1–518, 2-s2.0-79953877824, https://doi.org/10.1007/978-3-642-18363-8_1, Zbl1222.46002.
- 11 Ružička M., Electrorheological Fluids: Modeling and Mathematical Theory, 2000, 1748, Springer, Berlin, Germany, Lecture Notes in Mathematics, https://doi.org/10.1007/BFb0104029, MR1810360.
- 12 Bendahmane M., Wittbold P., and Zimmermann A., Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1-data, Journal of Differential Equations. (2010) 249, no. 6, 1483–1515, https://doi.org/10.1016/j.jde.2010.05.011, MR2677803.
- 13 Akagi G. and Matsuura K., Well-posedness and large-time behaviors of solutions for a parabolic equation involving p(x)-Laplacian, Discrete and Continuous Dynamical Systems - Series A, Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl. Vol. I. (2011) 22–31, MR3012130.
- 14 Edmunds D. E. and Rakosnik J., Sobolev embeddings with variable exponent, Studia Mathematica. (2000) 143, no. 3, 267–293, MR1815935.
- 15 Edmunds D. E. and Rakosnik J., Sobolev embeddings with variable exponent, II, Mathematische Nachrichten. (2002) 246/247, 53–67, https://doi.org/10.1002/1522-2616(200212)246:1<53::AID-MANA53>3.0.CO;2-T, MR1944549.
- 16 Fan X. and Zhao D., On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), Journal of Mathematical Analysis and Applications. (2001) 263, no. 2, 424–446, https://doi.org/10.1006/jmaa.2000.7617, MR1866056.
- 17 Simon L., Application of Monotone Type Operators to Nonlinear PDEs, 2013, Prime Rate Kft., Budapest, Hungary.
- 18
Quarteroni A. and
Valli A., Numerical approximation of partial differential equations, 1994, 23, Springer-Verlag, Berlin, Germany, Springer Series in Computational Mathematics, MR1299729.
10.1007/978-3-540-85268-1 Google Scholar
- 19 Li D. and Wang J., Unconditionally optimal error analysis of Crank-Nicolson Galerkin {FEM}s for a strongly nonlinear parabolic system, Journal of Scientific Computing. (2017) 72, no. 2, 892–915, MR3673699, https://doi.org/10.1007/s10915-017-0381-3, Zbl06805231, 2-s2.0-85012863881.
- 20
Leray J. and
Lions J.-L., Quelques résulatats de Visick sur les problémes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bulletin de la Société Mathématique de France. (1965) 93, 97–107, MR0194733.
10.24033/bsmf.1617 Google Scholar
- 21 Hecht F., Freefem++ manual, http, http://www.freefem.org.
