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February  2014, 7(1): 1-16. doi: 10.3934/dcdss.2014.7.1

Doubly nonlinear parabolic equations involving variable exponents

Goro Akagi 1,

1. 

Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan

Received  February 2012 Revised  January 2013 Published  July 2013

This paper is concerned with doubly nonlinear parabolic equations involving variable exponents. The existence of solutions is proved by developing an abstract theory on doubly nonlinear evolution equations governed by gradient operators. In contrast to constant exponent cases, two nonlinear terms have inhomogeneous growth and some difficulty may occur in establishing energy estimates. Our method of proof relies on an efficient use of Legendre-Fenchel transforms of convex functionals and an energy method.
Keywords: subdifferential, parabolic, variable exponent, $p(x)$-Laplacian, evolution equation., Doubly nonlinear.
Mathematics Subject Classification: Primary: 35K92, 47J35; Secondary: 46E3.
Citation: Goro Akagi. Doubly nonlinear parabolic equations involving variable exponents. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 1-16. doi: 10.3934/dcdss.2014.7.1
References:
[1]

E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., 156 (2001), 121-140. doi: 10.1007/s002050100117.

[2]

G. Akagi, Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces, J. Differential Equations, 231 (2006), 32-56. doi: 10.1016/j.jde.2006.04.006.

[3]

G. Akagi and K. Matsuura, Nonlinear diffusion equations driven by the $p(\cdot)$-Laplacian, Nonlinear Differential Equations and Applications (NoDEA), 20 (2013), 37-64. doi: 10.1007/s00030-012-0153-6.

[4]

G. Akagi, Energy solutions of the Cauchy-Neumann problem for porous medium equations, Discrete and Continuous Dynamical Systems, Dynamical Systems, Differential Equations and Applications. $7^{th}$th AIMS Conference, suppl., (2009), 1-10.

[5]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474.

[6]

S. Antontsev and S. Shmarev, Parabolic equations with double variable nonlinearities, Math. Comput. Simulation, 81 (2011), 2018-2032. doi: 10.1016/j.matcom.2010.12.015.

[7]

S. Antontsev and S. Shmarev, Extinction of solutions of parabolic equations with variable anisotropic nonlinearities, Proc. Steklov Inst. Math., 261 (2008), 11-21. doi: 10.1134/S0081543808020028.

[8]

S. N. Antontsev and S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal., 60 (2005), 515-545. doi: 10.1016/j.na.2004.09.026.

[9]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'' Noordhoff, Leiden, 1976.

[10]

V. Barbu, Existence for nonlinear Volterra equations in Hilbert spaces, SIAM J. Math. Anal., 10 (1979), 552-569. doi: 10.1137/0510052.

[11]

M. Bendahmane, P. Wittbold and A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and $L^1$-data, J. Differential Equations, 249 (2010), 1483-1515. doi: 10.1016/j.jde.2010.05.011.

[12]

T. M. Bokalo and O. M. Buhrii, Doubly nonlinear parabolic equations with variable exponents of nonlinearity, Ukrainian Math. J., 63 (2011), 709-728. doi: 10.1007/s11253-011-0537-5.

[13]

H. Brézis, "Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,'' Math Studies, Vol. 5, North-Holland, Amsterdam/New York, 1973.

[14]

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.

[15]

E. DiBenedetto and R. E. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal., 12 (1981), 731-751. doi: 10.1137/0512062.

[16]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, "Lebesgue and Sobolev Spaces with Variable Exponents,'' Lecture Notes in Mathematics, Vol. 2017, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[17]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446. doi: 10.1006/jmaa.2000.7617.

[18]

X. Fan, Y. Zhao and D. Zhao, Compact imbedding theorems with symmetry of Strauss-Lions type for the space $W^{1,p(x)}(\Omega)$,( J. Math. Anal. Appl., 255 (2001), 333-348. doi: 10.1006/jmaa.2000.7266.

[19]

Y. Fu and N. Pan, Existence of solutions for nonlinear parabolic problem with $p(x)$-growth, J. Math. Anal. Appl., 362 (2010), 313-326. doi: 10.1016/j.jmaa.2009.08.038.

[20]

O. Grange and F. Mignot, Sur la résolution d'une équation et d'une inéquation paraboliques non linéaires, J. Functional Analysis, 11 (1972), 77-92. doi: 10.1016/0022-1236(72)90080-8.

[21]

P. Harjulehto, P. Hästö, Ú.-V. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574. doi: 10.1016/j.na.2010.02.033.

[22]

N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975), 304-331. doi: 10.1007/BF02761596.

[23]

N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with time-dependent constraints, Nonlinear Anal., 10 (1986), 1181-1202. doi: 10.1016/0362-546X(86)90058-1.

[24]

K. Kurata and N. Shioji, Compact embedding from $W^{1,2}_0(\Omega)$ to $L^{q(x)}(\Omega)$ and its application to nonlinear elliptic boundary value problem with variable critical exponent, J. Math. Anal. Appl., 339 (2008), 1386-1394. doi: 10.1016/j.jmaa.2007.07.083.

[25]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires,'' Dunod, Paris, 1969.

[26]

J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications I,'' Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972.

[27]

E. Maitre and P. Witomski, A pseudo-monotonicity adapted to doubly nonlinear elliptic-parabolic equations, Nonlinear Anal., 50 (2002), 223-250. doi: 10.1016/S0362-546X(01)00748-9.

[28]

Y. Mizuta, T. Ohno, T. Shimomura and N. Shioji, Compact embeddings for Sobolev spaces of variable exponents and existence of solutions for nonlinear elliptic problems involving the $p(x)$-Laplacian and its critical exponent, Ann. Acad. Sci. Fenn. Math., 35 (2010), 115-130. doi: 10.5186/aasfm.2010.3507.

[29]

J. Musielak, "Orlicz Spaces and Modular Spaces,'' Lecture Notes in Mathematics, Vol. 1034, Springer-Verlag, Berlin, 1983.

[30]

P. A. Raviart, Sur la résolution de certaines équations paraboliques non linéaires, J. Functional Analysis, 5 (1970), 299-328. doi: 10.1016/0022-1236(70)90031-5.

[31]

T. Roubíček, "Nonlinear Partial Differential Equations with Applications,'' International Series of Numerical Mathematics, Vol. 153. Birkhäuser Verlag, Basel, 2005.

[32]

M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory,'' Lecture Notes in Mathematics, Vol. 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[33]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura. Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[34]

M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl., 132 (1988) 187-212. doi: 10.1016/0022-247X(88)90053-4.

[35]

N. Yamazaki, Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems, Discrete and Continuous Dynamical Systems, suppl., (2005), 920-929.

[36]

C. Zhang and S. Zhou, Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and $L^1$ data, J. Differential Equations, 248 (2010), 1376-1400. doi: 10.1016/j.jde.2009.11.024.

[37]

V. V. Zhikov, On the technique for passing to the limit in nonlinear elliptic equations, Functional Analysis and Its Applications, 43 (2009), 96-112. doi: 10.1007/s10688-009-0014-1.

show all references

References:
[1]

E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., 156 (2001), 121-140. doi: 10.1007/s002050100117.

[2]

G. Akagi, Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces, J. Differential Equations, 231 (2006), 32-56. doi: 10.1016/j.jde.2006.04.006.

[3]

G. Akagi and K. Matsuura, Nonlinear diffusion equations driven by the $p(\cdot)$-Laplacian, Nonlinear Differential Equations and Applications (NoDEA), 20 (2013), 37-64. doi: 10.1007/s00030-012-0153-6.

[4]

G. Akagi, Energy solutions of the Cauchy-Neumann problem for porous medium equations, Discrete and Continuous Dynamical Systems, Dynamical Systems, Differential Equations and Applications. $7^{th}$th AIMS Conference, suppl., (2009), 1-10.

[5]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474.

[6]

S. Antontsev and S. Shmarev, Parabolic equations with double variable nonlinearities, Math. Comput. Simulation, 81 (2011), 2018-2032. doi: 10.1016/j.matcom.2010.12.015.

[7]

S. Antontsev and S. Shmarev, Extinction of solutions of parabolic equations with variable anisotropic nonlinearities, Proc. Steklov Inst. Math., 261 (2008), 11-21. doi: 10.1134/S0081543808020028.

[8]

S. N. Antontsev and S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal., 60 (2005), 515-545. doi: 10.1016/j.na.2004.09.026.

[9]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'' Noordhoff, Leiden, 1976.

[10]

V. Barbu, Existence for nonlinear Volterra equations in Hilbert spaces, SIAM J. Math. Anal., 10 (1979), 552-569. doi: 10.1137/0510052.

[11]

M. Bendahmane, P. Wittbold and A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and $L^1$-data, J. Differential Equations, 249 (2010), 1483-1515. doi: 10.1016/j.jde.2010.05.011.

[12]

T. M. Bokalo and O. M. Buhrii, Doubly nonlinear parabolic equations with variable exponents of nonlinearity, Ukrainian Math. J., 63 (2011), 709-728. doi: 10.1007/s11253-011-0537-5.

[13]

H. Brézis, "Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,'' Math Studies, Vol. 5, North-Holland, Amsterdam/New York, 1973.

[14]

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.

[15]

E. DiBenedetto and R. E. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal., 12 (1981), 731-751. doi: 10.1137/0512062.

[16]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, "Lebesgue and Sobolev Spaces with Variable Exponents,'' Lecture Notes in Mathematics, Vol. 2017, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[17]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446. doi: 10.1006/jmaa.2000.7617.

[18]

X. Fan, Y. Zhao and D. Zhao, Compact imbedding theorems with symmetry of Strauss-Lions type for the space $W^{1,p(x)}(\Omega)$,( J. Math. Anal. Appl., 255 (2001), 333-348. doi: 10.1006/jmaa.2000.7266.

[19]

Y. Fu and N. Pan, Existence of solutions for nonlinear parabolic problem with $p(x)$-growth, J. Math. Anal. Appl., 362 (2010), 313-326. doi: 10.1016/j.jmaa.2009.08.038.

[20]

O. Grange and F. Mignot, Sur la résolution d'une équation et d'une inéquation paraboliques non linéaires, J. Functional Analysis, 11 (1972), 77-92. doi: 10.1016/0022-1236(72)90080-8.

[21]

P. Harjulehto, P. Hästö, Ú.-V. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574. doi: 10.1016/j.na.2010.02.033.

[22]

N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975), 304-331. doi: 10.1007/BF02761596.

[23]

N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with time-dependent constraints, Nonlinear Anal., 10 (1986), 1181-1202. doi: 10.1016/0362-546X(86)90058-1.

[24]

K. Kurata and N. Shioji, Compact embedding from $W^{1,2}_0(\Omega)$ to $L^{q(x)}(\Omega)$ and its application to nonlinear elliptic boundary value problem with variable critical exponent, J. Math. Anal. Appl., 339 (2008), 1386-1394. doi: 10.1016/j.jmaa.2007.07.083.

[25]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires,'' Dunod, Paris, 1969.

[26]

J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications I,'' Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972.

[27]

E. Maitre and P. Witomski, A pseudo-monotonicity adapted to doubly nonlinear elliptic-parabolic equations, Nonlinear Anal., 50 (2002), 223-250. doi: 10.1016/S0362-546X(01)00748-9.

[28]

Y. Mizuta, T. Ohno, T. Shimomura and N. Shioji, Compact embeddings for Sobolev spaces of variable exponents and existence of solutions for nonlinear elliptic problems involving the $p(x)$-Laplacian and its critical exponent, Ann. Acad. Sci. Fenn. Math., 35 (2010), 115-130. doi: 10.5186/aasfm.2010.3507.

[29]

J. Musielak, "Orlicz Spaces and Modular Spaces,'' Lecture Notes in Mathematics, Vol. 1034, Springer-Verlag, Berlin, 1983.

[30]

P. A. Raviart, Sur la résolution de certaines équations paraboliques non linéaires, J. Functional Analysis, 5 (1970), 299-328. doi: 10.1016/0022-1236(70)90031-5.

[31]

T. Roubíček, "Nonlinear Partial Differential Equations with Applications,'' International Series of Numerical Mathematics, Vol. 153. Birkhäuser Verlag, Basel, 2005.

[32]

M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory,'' Lecture Notes in Mathematics, Vol. 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[33]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura. Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[34]

M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl., 132 (1988) 187-212. doi: 10.1016/0022-247X(88)90053-4.

[35]

N. Yamazaki, Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems, Discrete and Continuous Dynamical Systems, suppl., (2005), 920-929.

[36]

C. Zhang and S. Zhou, Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and $L^1$ data, J. Differential Equations, 248 (2010), 1376-1400. doi: 10.1016/j.jde.2009.11.024.

[37]

V. V. Zhikov, On the technique for passing to the limit in nonlinear elliptic equations, Functional Analysis and Its Applications, 43 (2009), 96-112. doi: 10.1007/s10688-009-0014-1.

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