Existence of Weak Solutions for a Class of Quasilinear Parabolic Problems in Weighted Sobolev Space


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 Galerkins 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 T0.

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M. Qiu and L. Mei, "Existence of Weak Solutions for a Class of Quasilinear Parabolic Problems in Weighted Sobolev Space," Advances in Pure Mathematics, Vol. 3 No. 1A, 2013, pp. 204-208. doi: 10.4236/apm.2013.31A028.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] 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
[2] 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
[3] 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.
[4] 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
[5] P. Drabek, A. kufner and F. Nicolosi, “Quasilinear Elliptic Equations with Degenerations and Singularities,” Walter de Gruyter, Berlin, 1997. doi:10.1515/9783110804775
[6] 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.
[7] R. A. Adams and J. F. F. John, “Sobolev Space,” Academy Press, New York, 2009.
[8] M. Renardy and R. Rogers, “An Introduction to Partial Differential Equations,” Springer. New York, 2004.
[9] L. Evans, “Partial Differential Equations,” American Mathematical Society, Providence, 1998.
[10] A. Antonio, “On Compact Imbedding Theorems in Weighted Sobolev Spaces,” Czechoslovak Mathematical Journal, Vol. 104, No. 29, 1979, pp. 635-648.
[11] 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
[12] 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
[13] M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information,” Cambridge University Press, Cambridge, 2000.

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