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

Affiliation(s)

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China.

Center for Computational Geosciences, Xi’an Jiaotong University, Xi’an, China.

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China.

Center for Computational Geosciences, Xi’an Jiaotong University, Xi’an, China.

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.

KEYWORDS

Weighted Sobolev Space; Energy Estimates; Compact Imbedding; Sobolev Interpolation Inequalities

Weighted Sobolev Space; Energy Estimates; Compact Imbedding; Sobolev Interpolation Inequalities

Cite this paper

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. 1, 2013, pp. 204-208. doi: 10.4236/apm.2013.31A028.

M. Qiu and L. Mei, "Existence of Weak Solutions for a Class of Quasilinear Parabolic Problems in Weighted Sobolev Space,"

References

[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.

[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.