Journal Menu
- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 312536, 17 pages
doi:10.1155/2012/312536
Research Article
Asymptotic Behavior of Approximated Solutions to Parabolic Equations with Irregular Data
1School of Mathematical Sciences, Anhui University, Hefei 230039, China
2School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China
Received 10 July 2012; Accepted 17 August 2012
Academic Editor: Sergey V. Zelik
Copyright © 2012 Weisheng Niu and Hongtao Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- E. Casas, “Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations,” SIAM Journal on Control and Optimization, vol. 35, no. 4, pp. 1297–1327, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- T. Goudon and M. Saad, “On a Fokker-Planck equation arising in population dynamics,” Revista Matemática Complutense, vol. 11, no. 2, pp. 353–372, 1998. View at Zentralblatt MATH
- R. Lewandowski, “The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 28, no. 2, pp. 393–417, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- P.-L. Lions, Mathematical Topics in Fluid Mechanics. Volume 1 Incompressible Models, vol. 3 of Oxford Lecture Series in Mathematics and Its Applications, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, NY, USA, 1996.
- H. Amann and P. Quittner, “Semilinear parabolic equations involving measures and low regularity data,” Transactions of the American Mathematical Society, vol. 356, no. 3, pp. 1045–1119, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- F. Andreu, J. M. Mazon, S. S. de León, and J. Toledo, “Existence and uniqueness for a degenerate parabolic equation with -data,” Transactions of the American Mathematical Society, vol. 351, no. 1, pp. 285–306, 1999. View at Publisher · View at Google Scholar
- L. Boccardo and T. Gallouët, “Nonlinear elliptic and parabolic equations involving measure data,” Journal of Functional Analysis, vol. 87, no. 1, pp. 149–169, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. Prignet, “Existence and uniqueness of “entropy” solutions of parabolic problems with data,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 28, no. 12, pp. 1943–1954, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. Dall'Aglio, “Approximated solutions of equations with data. Application to the -convergence of quasi-linear parabolic equations,” Annali di Matematica Pura ed Applicata, vol. 170, no. 4, pp. 207–240, 1996. View at Publisher · View at Google Scholar
- D. Blanchard and F. Murat, “Renormalised solutions of nonlinear parabolic problems with data: existence and uniqueness,” Proceedings of the Royal Society of Edinburgh A, vol. 127, no. 6, pp. 1137–1152, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- M. Efendiev and S. Zelik, “Finite- and infinite-dimensional attractors for porous media equations,” Proceedings of the London Mathematical Society. 3rd series, vol. 96, no. 1, pp. 51–77, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- M. Nakao and N. Aris, “On global attractor for nonlinear parabolic equations of -Laplacian type,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 793–809, 2007. View at Publisher · View at Google Scholar
- F. Petitta, “Asymptotic behavior of solutions for linear parabolic equations with general measure data,” Comptes Rendus Mathématique I, vol. 344, no. 9, pp. 571–576, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- F. Petitta, “Asymptotic behavior of solutions for parabolic operators of Leray-Lions type and measure data,” Advances in Differential Equations, vol. 12, no. 8, pp. 867–891, 2007. View at Zentralblatt MATH
- C. Zhong and W. Niu, “Long-time behavior of solutions to nonlinear reaction diffusion equations involving data,” Communications in Contemporary Mathematics, vol. 14, no. 1, Article ID 1250007, 19 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- W. Niu and C. Zhong, “Global attractors for the p-Laplacian equations with nonregular data,” Journal of Mathematical Analysis and Applications, vol. 393, pp. 56–65, 2012. View at Publisher · View at Google Scholar
- F. Petitta, “A non-existence result for nonlinear parabolic equations with singular measures as data,” Proceedings of the Royal Society of Edinburgh A, vol. 139, no. 2, pp. 381–392, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- J. Droniou, A. Porretta, and A. Prignet, “Parabolic capacity and soft measures for nonlinear equations,” Potential Analysis, vol. 19, no. 2, pp. 99–161, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- L. Boccardo, T. Gallouët, and L. Orsina, “Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data,” Annales de l'Institut Henri Poincaré, vol. 13, no. 5, pp. 539–551, 1996. View at Zentralblatt MATH
- L. Boccardo, T. Gallouët, and J. L. Vázquez, “Nonlinear elliptic equations in RN without growth restrictions on the data,” Journal of Differential Equations, vol. 105, no. 2, pp. 334–363, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. Prignet, “Remarks on existence and uniqueness of solutions of elliptic problems with right-hand side measures,” Rendiconti di Matematica e delle sue Applicazioni, vol. 15, no. 3, pp. 321–337, 1995. View at Zentralblatt MATH
- F. Petitta, “Renormalized solutions of nonlinear parabolic equations with general measure data,” Annali di Matematica Pura ed Applicata, vol. 187, no. 4, pp. 563–604, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction To Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, UK, 2001.
- J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, UK, 2000. View at Publisher · View at Google Scholar
- Q. Ma, S. Wang, and C. Zhong, “Necessary and sufficient conditions for the existence of global attractors for semigroups and applications,” Indiana University Mathematics Journal, vol. 51, no. 6, pp. 1541–1559, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, NY, USA, 1997.
- C.-K. Zhong, M.-H. Yang, and C.-Y. Sun, “The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations,” Journal of Differential Equations, vol. 223, no. 2, pp. 367–399, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- C. Sun, “Asymptotic regularity for some dissipative equations,” Journal of Differential Equations, vol. 248, no. 2, pp. 342–362, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- L. Boccardo, T. Gallouët, and J. L. Vazquez, “Solutions of nonlinear parabolic equations without growth restrictions on the data,” Electronic Journal of Differential Equations, vol. 2001, no. 60, pp. 1–20, 2001. View at Zentralblatt MATH