Research ArticleOpen AccessGoogle Scholar indexed
Existence of Weak Solutions for a Class of Quasilinear Parabolic Problems in Weighted Sobolev Space
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China
Center for Computational Geosciences, Xi’an Jiaotong University, Xi’an, China
- 1 School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China
- 2 Center for Computational Geosciences, Xi’an Jiaotong University, Xi’an, China
Advances in Pure Mathematics·Volume 03 (2013)·Pages 204–208·Published 30 January 2013·DOI10.4236/apm.2013.31A028
Copy link · social · email
Abstract
In this paper, we investigate the existence and uniqueness of weak solutions for a new class of initial/boundary-value parabolic problems with nonlinear perturbation term in weighted Sobolev space. By building up the compact imbedding in weighted Sobolev space and extending Galerkin ’ s method to a new class of nonlinear problems, we drive out that there exists at least one weak solution of the nonlinear equations in the interval [ 0,T ] for the fixed time T ＞ 0 .
KeywordsWeighted Sobolev SpaceEnergy EstimatesCompact ImbeddingSobolev Interpolation Inequalities
- C. O. Alvesa and A. El Hamidib, “Nehari Manifold and Existence of Positive Solutions to a Class of Quasilinear Problems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 60, No. 4, 2005, pp. 611-624. doi:10.1016/j.na.2004.09.039
- K. J. Brown and Y. P. Zhang, “The Nehari Manifold for a Semilinear Elliptic Problem with a Sign-Changing Weight Function,” Journal of Differential Equations, Vol. 193, No. 2, 2003, pp. 481-499. doi:10.1016/S0022-0396(03)00121-9
- J. Huang and Z. L. Pu, “The Nehari Manifold of Nonlinear Elliptic Equations,” Journal of Sichuan Normal University, Vol. 31, No. 2, 2007, pp. 18-32.
- T. Bartsch and M. Willem, “On an Elliptic Equation with Concave and Convex Nonlinearities,” Proceedings of the American Mathematical Society, Vol. 123, 1995, pp. 3555-3561. doi:10.1090/S0002-9939-1995-1301008-2
- P. Drabek, A. kufner and F. Nicolosi, “Quasilinear Elliptic Equations with Degenerations and Singularities,” Walter de Gruyter, Berlin, 1997. doi:10.1515/9783110804775
- P. A. Binding, P. Drabek and Y. X. Huang, “On Neumann Boundary Value Problems for Some Quasilinear Elliptic Equations,” Electronic Journal of Differential Equations, Vol. 1997, No. 5, 1997, pp. 1-11.
- R. A. Adams and J. F. F. John, “Sobolev Space,” Academy Press, New York, 2009.
- M. Renardy and R. Rogers, “An Introduction to Partial Differential Equations,” Springer. New York, 2004.
- L. Evans, “Partial Differential Equations,” American Mathematical Society, Providence, 1998.
- A. Antonio, “On Compact Imbedding Theorems in Weighted Sobolev Spaces,” Czechoslovak Mathematical Journal, Vol. 104, No. 29, 1979, pp. 635-648.
- T. F. Wu, “On Semilinear Elliptic Equations Involving Concave-Convex Nonlinearities and Sign-Changing Weight Function,” Journal of Mathematical Analysis and Applications, Vol. 318, No. 1, 2006, pp. 253-270. doi:10.1016/j.jmaa.2005.05.057
- M. L. Miotto and O. H. Miyagaki, “Multiple Positive Solutions for Semilinear Dirichlet Problems with Sign-Changing Weight Function in Infinite Strip Domains,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 7-8, 2009, pp. 3434-3447. doi:10.1016/j.na.2009.02.010
- M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information,” Cambridge University Press, Cambridge, 2000.