American Institute of Mathematical Sciences

Advanced
  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.

[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.

[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. 

[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.

[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.

[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. 

[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.

[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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