American Institute of Mathematical Sciences

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  2017, 24: 38-52. doi: 10.3934/era.2017.24.005

Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations

Peiying Chen

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received  December 16, 2016 Revised  April 30, 2017 Published  June 2017

Fund Project: The author would like to thank her supervisor Prof. Zhongrui Shi, who supported her throughout her paper with his knowledge, patience and excellent guidance.

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.

Keywords: Existence, uniqueness, weak solution, variation problem, $\mathcal{N}$-function.
Mathematics Subject Classification: 35K55(Primary), 35K61(Secondary), 46B20.
Citation: Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005
References:
[1]

G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer-Verlag, New York, 2002.  Google Scholar

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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.  doi: 10.1080/16073606.1994.9631762.  Google Scholar

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J. M. Ball and F. Murat, Remarks on Chacon's biting lemma, Proc. Amer. Math. Soc., 107 (1989), 655-663.  doi: 10.2307/2048162.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. Fuchs and V. Osmolovski, Variational integrals on Orlicz-Sobolev spaces, Z. Anal. Anwendungen, 17 (1998), 393-415.  doi: 10.4171/ZAA/829.  Google Scholar

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N. Fukagai and K. Narukawa, Nonlinear eigenvalue problem for a model equation of an elastic surface, Hiroshima Math. J., 25 (1995), 19-41.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Marcel Dekker, Inc., New York, 2002. doi: 10.1201/9780203910863.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer-Verlag, New York, 2002.  Google Scholar

[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.  Google Scholar

[3]

R. Adams, Sobolev Spaces, Academic Press, New York-London, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Fuchs and V. Osmolovski, Variational integrals on Orlicz-Sobolev spaces, Z. Anal. Anwendungen, 17 (1998), 393-415.  doi: 10.4171/ZAA/829.  Google Scholar

[12]

N. Fukagai and K. Narukawa, Nonlinear eigenvalue problem for a model equation of an elastic surface, Hiroshima Math. J., 25 (1995), 19-41.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Marcel Dekker, Inc., New York, 2002. doi: 10.1201/9780203910863.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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