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Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations
Department of Mathematics, Shanghai University, Shanghai 200444, China |
In this paper, we study the Dirichlet boundary value problem of a class of nonlinear parabolic equations. By a priori estimates, difference and variation techniques, we establish the existence and uniqueness of weak solutions of this problem.
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G. Aubert and P. Kornprobst,
Mathematical Problems in Image Processing, Springer-Verlag, New York, 2002. |
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J. Alexopoulos,
de la Vallée Poussin's theorem and weakly compact sets in Orlicz spaces, Quaestiones Math., 17 (1994), 231-248.
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Sobolev Spaces, Academic Press, New York-London, 1975. |
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Remarks on Chacon's biting lemma, Proc. Amer. Math. Soc., 107 (1989), 655-663.
doi: 10.2307/2048162. |
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P. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt,
Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations, 11 (2000), 33-62.
doi: 10.1007/s005260050002. |
[6] |
Y. Chen, S. Levine and M. Rao,
Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
[7] |
L. Diening, Theoerical and Numerical Results for Electrorheological Fluids, Ph. D. Thesis, University of Freiburg, Germany, 2002. Google Scholar |
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L. C. Evans,
Weak Convergence Methods for Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1990.
doi: 10.1090/cbms/074. |
[9] |
G. Fragnelli,
Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl., 367 (2010), 204-228.
doi: 10.1016/j.jmaa.2009.12.039. |
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M. Fuchs and L. Gongbao,
Variational inequalities for energy functionals with nonstandard growth conditions, Abstr. Appl. Anal., 3 (1998), 41-64.
doi: 10.1155/S1085337598000438. |
[11] |
M. Fuchs and V. Osmolovski,
Variational integrals on Orlicz-Sobolev spaces, Z. Anal. Anwendungen, 17 (1998), 393-415.
doi: 10.4171/ZAA/829. |
[12] |
N. Fukagai and K. Narukawa,
Nonlinear eigenvalue problem for a model equation of an elastic surface, Hiroshima Math. J., 25 (1995), 19-41.
|
[13] |
Z. Feng and Z. Yin,
On weak solutions for a class of nonlinear parabolic equations related to image analysis, Nonlinear Anal., 71 (2009), 2506-2517.
doi: 10.1016/j.na.2009.01.087. |
[14] |
P. Gwiazda and A. Świerczewska-Gwiazda,
On non-Newtonian fluids with a property of rapid thickeninig under different stimulus, Math. Models Methods Appl. Sci., 18 (2008), 1073-1092.
doi: 10.1142/S0218202508002954. |
[15] |
M. M. Rao and Z. D. Ren,
Applications of Orlicz Spaces, Marcel Dekker, Inc., New York, 2002.
doi: 10.1201/9780203910863. |
[16] |
K. R. Rajagopal and M. Ružička, Mathematical modelling of electrorheological fluids, Continuum Mech. Thermodyn., 13 (2001), 59-78. Google Scholar |
[17] |
M. Saadoune and M. Valadier,
Extraction of ''good" subsequence from a bounded sequence of integrable functions, J. Convex Anal., 2 (1995), 345-357.
|
[18] |
C. Wu, Convex Functions and Orlicz Spaces, Science Press, Beijing, 1961. Google Scholar |
[19] |
L. Wang and S. Zhou,
Existence and uniqueness of weak solutions for a nonlinear parabolic equation related to image analysis, J. Partial Differential Equations, 19 (2006), 97-112.
|
[20] |
V. V. Zhikov,
Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv., 9 (1987), 33-66.
doi: 10.1070/IM1987v029n01ABEH000958. |
show all references
References:
[1] |
G. Aubert and P. Kornprobst,
Mathematical Problems in Image Processing, Springer-Verlag, New York, 2002. |
[2] |
J. Alexopoulos,
de la Vallée Poussin's theorem and weakly compact sets in Orlicz spaces, Quaestiones Math., 17 (1994), 231-248.
doi: 10.1080/16073606.1994.9631762. |
[3] |
R. Adams,
Sobolev Spaces, Academic Press, New York-London, 1975. |
[4] |
J. M. Ball and F. Murat,
Remarks on Chacon's biting lemma, Proc. Amer. Math. Soc., 107 (1989), 655-663.
doi: 10.2307/2048162. |
[5] |
P. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt,
Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations, 11 (2000), 33-62.
doi: 10.1007/s005260050002. |
[6] |
Y. Chen, S. Levine and M. Rao,
Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
[7] |
L. Diening, Theoerical and Numerical Results for Electrorheological Fluids, Ph. D. Thesis, University of Freiburg, Germany, 2002. Google Scholar |
[8] |
L. C. Evans,
Weak Convergence Methods for Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1990.
doi: 10.1090/cbms/074. |
[9] |
G. Fragnelli,
Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl., 367 (2010), 204-228.
doi: 10.1016/j.jmaa.2009.12.039. |
[10] |
M. Fuchs and L. Gongbao,
Variational inequalities for energy functionals with nonstandard growth conditions, Abstr. Appl. Anal., 3 (1998), 41-64.
doi: 10.1155/S1085337598000438. |
[11] |
M. Fuchs and V. Osmolovski,
Variational integrals on Orlicz-Sobolev spaces, Z. Anal. Anwendungen, 17 (1998), 393-415.
doi: 10.4171/ZAA/829. |
[12] |
N. Fukagai and K. Narukawa,
Nonlinear eigenvalue problem for a model equation of an elastic surface, Hiroshima Math. J., 25 (1995), 19-41.
|
[13] |
Z. Feng and Z. Yin,
On weak solutions for a class of nonlinear parabolic equations related to image analysis, Nonlinear Anal., 71 (2009), 2506-2517.
doi: 10.1016/j.na.2009.01.087. |
[14] |
P. Gwiazda and A. Świerczewska-Gwiazda,
On non-Newtonian fluids with a property of rapid thickeninig under different stimulus, Math. Models Methods Appl. Sci., 18 (2008), 1073-1092.
doi: 10.1142/S0218202508002954. |
[15] |
M. M. Rao and Z. D. Ren,
Applications of Orlicz Spaces, Marcel Dekker, Inc., New York, 2002.
doi: 10.1201/9780203910863. |
[16] |
K. R. Rajagopal and M. Ružička, Mathematical modelling of electrorheological fluids, Continuum Mech. Thermodyn., 13 (2001), 59-78. Google Scholar |
[17] |
M. Saadoune and M. Valadier,
Extraction of ''good" subsequence from a bounded sequence of integrable functions, J. Convex Anal., 2 (1995), 345-357.
|
[18] |
C. Wu, Convex Functions and Orlicz Spaces, Science Press, Beijing, 1961. Google Scholar |
[19] |
L. Wang and S. Zhou,
Existence and uniqueness of weak solutions for a nonlinear parabolic equation related to image analysis, J. Partial Differential Equations, 19 (2006), 97-112.
|
[20] |
V. V. Zhikov,
Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv., 9 (1987), 33-66.
doi: 10.1070/IM1987v029n01ABEH000958. |
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