# American Institue of Mathematical Sciences

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

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

 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.

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. [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, 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, 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, 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. [8] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations Amer. Math. Soc. , Providence, RI, 1990. [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, 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, V. Osmolovski, Variational integrals on Orlicz-Sobolev spaces, Z. Anal. Anwendungen, 17 (1998), 393-415. doi: 10.4171/ZAA/829. [12] N. Fukagai, K. Narukawa, Nonlinear eigenvalue problem for a model equation of an elastic surface, Hiroshima Math. J., 25 (1995), 19-41. [13] Z. Feng, 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, 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. [16] K. R. Rajagopal, M. Ružička, Mathematical modelling of electrorheological fluids, Continuum Mech. Thermodyn., 13 (2001), 59-78. [17] M. Saadoune, 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. [19] L. Wang, 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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##### 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, 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, 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, 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. [8] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations Amer. Math. Soc. , Providence, RI, 1990. [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, 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, V. Osmolovski, Variational integrals on Orlicz-Sobolev spaces, Z. Anal. Anwendungen, 17 (1998), 393-415. doi: 10.4171/ZAA/829. [12] N. Fukagai, K. Narukawa, Nonlinear eigenvalue problem for a model equation of an elastic surface, Hiroshima Math. J., 25 (1995), 19-41. [13] Z. Feng, 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, 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. [16] K. R. Rajagopal, M. Ružička, Mathematical modelling of electrorheological fluids, Continuum Mech. Thermodyn., 13 (2001), 59-78. [17] M. Saadoune, 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. [19] L. Wang, 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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