Existence and uniqueness of weak solutions for a class of nonlinear divergence type diffusion equations

Document Type : Research Paper

Author

Department of Mathematics‎, ‎Shanghai University‎, ‎Shanghai 200444‎, ‎China.

Abstract

‎In this paper‎, ‎we study the Neumann boundary value problem of a class of nonlinear divergence type diffusion equations‎. ‎By a priori estimates‎, ‎difference and variation techniques‎, ‎we establish the existence and uniqueness of weak solutions of this problem.

Keywords

Main Subjects


R. Adams, Sobolev Spaces, Acard. Press, New York, 1975.
G. Aubert and P. Kornprost, Mathematical Problems in Image Processing, Springer-Verlag, Berlin, 2002.
J.M. Ball and F. Murat, Remarks on Chacon's biting lemma, Proc. Amer. Math. Soc. 107 (1989), no. 3, 655--663.
Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383--1406.
L. Diening, Theoerical and Numerical Results for Electrorheological Fluids, PhD Thesis, University of Freiburg, Germany, 2002.
L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Amer. Math. Soc. Providence, RI, 1990.
Zh. Feng and Zh. Yin, On weak solutions for a class of nonlinear parabolic equations related to image analysis, Nonlinear Anal. 71 (2009), no. 7, 2506--2517.
G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl. 367 (2010) 204--228.
M. Fuchs and G. Li, Variational inequalities for energy functionals with nonstandard growth conditions, Abstr. Appl. Anal. 3 (1998) 41--64.
M. Fuchs and V. Osmolovski, Variational integrals on Orlicz-Sobolev spaces, Z. Anal. Anwend. 17 (1998) 393--415.
N. Fukagai and K. Narukawa, Nonlinear eigenvalue problem for a model equation of an elastic surface, Hiroshima Math. J. 25 (1995) 19--41.
P. Gwiazda and A. Swierczewska--Gwiazda, On non-newtonian uids with a property of rapid thickeninig under different stimulus, Math. Models Methods Appl. Sci. 18 (2008), no. 7, 1073--1092.
P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine Intelligence 12 (1990), no. 7, 629--639.
K.R. Rajagopal and M. Ruzicka, Mathematical modelling of electrorheological uids, Continuum Mech. Thermodynamics 13 (2001) 59--78.
M.M. Rao and Z.D. Ren, Applications of Orlicz Spaces, Marcel Dekker, New York, 2002.
M. Saadoune and M. Valadier, Extraction of good" subsequence from a bounded sequence of integrable functions, J. Convex Anal. 2 (1995) 345--357.
L. Wang and S. Zhou, Existence and uniqueness of weak solutions for a nonlinear parabolic equation related to image analysis, J. Partial Differential Equation 19 (2006), no. 2, 97--112.
C. Wu, Convex functions and Orlicz spaces, Science Press, BeiJing, 1961.
V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. 9 (1987) 33--66.