The Existence of Entropy Solutions for a Class of Parabolic Equations
Abstract
:1. Introduction
- ()
- for a.e. and any ;
- ()
- for a.e. and any ;
- ()
- for a.e. and all with ,
2. Notations, Definitions and Preliminary Results
3. Proof of the Weak Solution
4. Proof of the Entropy Solution
- (1)
- ,
- (2)
- ,
- (3)
- ,
- (4)
- ,
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Ruzicka, M. Electrorheological Fluids Modeling and Mathematical Theory; Springer: Berlin/Heidelberg, Germany, 2000. [Google Scholar]
- Sin, C. Boundary partial regularity for steady flows of electrorheological fluids in 3D bounded domains. Nonlinear Anal. 2019, 179, 309–343. [Google Scholar] [CrossRef]
- Theljani, A.; Belhachmi, Z.; Moakher, M. High-order anisotropic diffusion operators in spaces of variable exponents and application to image inpainting and restoration problems. Nonlinear Anal. Real World Appl. 2019, 47, 251–271. [Google Scholar] [CrossRef]
- Alves, C.O.; Simsen, J.; Simsen, M.S. Parabolic problems in ℝN with spatially variable exponents. Asymptot. Anal. 2015, 93, 51–64. [Google Scholar]
- Alves, C.O.; Shmarev, S.; Simsen, J.; Simsen, M.S. The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: Existence and asymptotic behavior. J. Math. Anal. Appl. 2016, 443, 265–294. [Google Scholar] [CrossRef]
- Bokalo, M.M.; Ilnytska, O.V. Problems for parabolic equations with variable exponents of nonlinearity and time delay. Appl. Anal. 2017, 96, 1240–1254. [Google Scholar] [CrossRef]
- Sabri, A.; Jamea, A.; Alaoui, H.T. Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and L1-data. Commun. Math. 2020, 28, 67–88. [Google Scholar] [CrossRef]
- Shangerganesh, L.; Gurusamy, A.; Balachandran, K. Weak solutions for nonlinear parabolic equations with variable exponents. Commun. Math. 2017, 25, 55–70. [Google Scholar] [CrossRef]
- Saintier, N.; Silva, A. Local existence conditions for an equations involving the p(x)-Laplacian with critical exponent in ℝN. Nonlinear Differ. Equ. Appl. 2017, 24, 19. [Google Scholar] [CrossRef]
- Xie, W.; Chen, H. Existence and multiplicity of solutions for p(x)-Laplacian equations in ℝN. Math. Nachr. 2018, 291, 2476–2488. [Google Scholar] [CrossRef]
- Youssfi, A.; Azroul, E.; Lahmi, B. Nonlinear parabolic equations with nonstandard growth. Appl. Anal. 2016, 95, 2766–2778. [Google Scholar] [CrossRef]
- Zhang, C.; Zhang, X. Renormalized solutions for the fractional p(x)-Laplacian equation with L1 data. Nonlinear Anal. 2020, 190, 111610. [Google Scholar] [CrossRef]
- DiPerna, R.J.; Lions, P.L. On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. 1989, 130, 321–366. [Google Scholar] [CrossRef]
- Bënilan, P.; Boccardo, L.; Gallouxext, T.; Gariepy, R.; Pierre, M.; Vazquez, J.L. An L1-theory of existence and uniqueness of nonlinear elliptic equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 1995, 22, 241–273. [Google Scholar]
- Azroul, E.; Benboubker, M.B.; Rhoudaf, M. On some p(x)-quasilinear problem with right-hand side measure. Math. Comput. Simul. 2014, 102, 117–130. [Google Scholar] [CrossRef]
- Bendahmane, M.; Wittbold, P. Renormalized solutions for nonlinear elliptic equations with variable exponents and L1 data. Nonlinear Anal. 2009, 70, 567–583. [Google Scholar] [CrossRef]
- Zhang, C.; Zhou, S. Entropy and renormalized solutions for the p(x)-Laplacian equation with measure data. Bull. Aust. Math. Soc. 2010, 82, 459–479. [Google Scholar] [CrossRef]
- Bendahmane, M.; Wittbold, P.; Zimmermann, A. Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1-data. J. Differ. Equ. 2010, 249, 1483–1515. [Google Scholar] [CrossRef]
- Chai, X.; Li, H.; Niu, W. Large time behavior for p(x)-Laplacian equations with irregular data. Electron. J. Differ. Equ. 2015, 2015, 1–16. [Google Scholar]
- Li, Z.; Gao, W. Existence of renormalized solutions to a nonlinear parabolic equation in L1 setting with nonstandard growth condition and gradient term. Math. Methods Appl. Sci. 2015, 38, 3043–3062. [Google Scholar] [CrossRef]
- Niu, W.; Chai, X. Global attractors for nonlinear parabolic equations with nonstandard growth and irregular data. J. Math. Anal. Appl. 2017, 451, 34–63. [Google Scholar]
- 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. [Google Scholar] [CrossRef]
- Kyelem, B.A.; Ouedraogo, A.; Zongo, F. Existence and uniqueness of entropy solutions to nonlinear parabolic problem with homogeneous Dirichlet boundary conditions involving variable exponent. SeMA J. 2019, 76, 153–180. [Google Scholar] [CrossRef]
- Mukminov, F.K. Uniqueness of the renormalized solution of an elliptic-parabolic problem in anisotropic Sobolev-Orlicz spaces. Sb. Math. 2017, 208, 1187–1206. [Google Scholar] [CrossRef]
- Mukminov, F.K. Existence of a renormalized solution to an anisotropic parabolic problem with variable nonlinearity exponents. Sb. Math. 2018, 209, 714–738. [Google Scholar] [CrossRef]
- Kozhevnikova, L.M. On the entropy solution to an elliptic problem in anisotropic Sobolev-Orlicz spaces. Comput. Math. Math. Phys. 2017, 57, 434–452. [Google Scholar] [CrossRef]
- Sanchón, M.; Urbano, J.M. Entropy solutions for the p(x)-Laplace equation. Trans. Am. Math. Soc. 2009, 361, 6387–6405. [Google Scholar] [CrossRef]
- Mokhtari, F.; Mecheter, R. Anisotropic degenerate parabolic problems in ℝN with variable exponent and locally integrable data. Mediterr. J. Math. 2019, 16, 1660–5454. [Google Scholar] [CrossRef]
- Teng, K.; Zhang, C.; Zhou, S. Renormalized and entropy solutions for the fractional p-Laplacian evolution equations. J. Evol. Equ. 2019, 19, 559–584. [Google Scholar] [CrossRef]
- Jamea, A.; Lamrani, A.A.; Hachimi, A.E. Existence of entropy solutions to nonlinear parabolic problems with variable exponent and L1-data. Ric. Mat. 2018, 67, 785–801. [Google Scholar] [CrossRef]
- Zhan, H. The entropy solution of a hyperbolic-parabolic mixed type equation. SpringerPlus 2016, 5, 1811–1824. [Google Scholar] [CrossRef]
- Diening, L.; Harjulehto, P.; Hästö, P.; Ruzicka, M. Lebesgue and Sobolev Spaces with Variable Exponents; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2011; Volume 2017. [Google Scholar]
- Fan, X.; Zhao, D. On the spaces Lp(x)(Ω) and Wm,p(x)(Ω). J. Math. Anal. Appl. 2001, 263, 424–446. [Google Scholar] [CrossRef]
- Azroul, E.; Benboubker, M.B.; Ouaro, S. Entropy solutions for nonlinear nonhomogeneous Neumann problems involving the generalized p(x)-Laplace operator. J. Appl. Anal. Comput. 2013, 3, 105–121. [Google Scholar] [CrossRef]
- Lions, J.L. Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires; Dunod: Paris, France, 1969. [Google Scholar]
- Babin, A.V.; Vishik, M.I. Attractors of partial differential evolution equations in an unbounded domain. Proc. R. Soc. Edinb. 1990, 116A, 221–243. [Google Scholar] [CrossRef]
- Qian, C.; Shen, Z. Existence of global solutions and attractors for the parabolic equation with critical Sobolev and Hardy exponent in ℝN. Nonlinear Anal. Real World Appl. 2018, 42, 290–307. [Google Scholar] [CrossRef]
- Landes, R. On the existence of weak solutions for quasilinear parabolic initial boundary value problems. Proc. R. Soc. Edinb. Sect. A 1981, 89, 217–237. [Google Scholar] [CrossRef]
- Porretta, A. Existence results for nonlinear parabolic equations via strong convergence of truncations. Ann. Mat. Pura Appl. 1999, 177, 143–172. [Google Scholar] [CrossRef]
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Chen, Z.; Shen, B. The Existence of Entropy Solutions for a Class of Parabolic Equations. Mathematics 2023, 11, 3753. https://doi.org/10.3390/math11173753
Chen Z, Shen B. The Existence of Entropy Solutions for a Class of Parabolic Equations. Mathematics. 2023; 11(17):3753. https://doi.org/10.3390/math11173753
Chicago/Turabian StyleChen, Zengfei, and Bingliang Shen. 2023. "The Existence of Entropy Solutions for a Class of Parabolic Equations" Mathematics 11, no. 17: 3753. https://doi.org/10.3390/math11173753
APA StyleChen, Z., & Shen, B. (2023). The Existence of Entropy Solutions for a Class of Parabolic Equations. Mathematics, 11(17), 3753. https://doi.org/10.3390/math11173753