# American Institute of Mathematical Sciences

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

## Doubly nonlinear parabolic equations involving variable exponents

 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.
Citation: Goro Akagi. Doubly nonlinear parabolic equations involving variable exponents. Discrete & 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. doi: 10.1007/s002050100117. Google Scholar [2] G. Akagi, Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces,, J. Differential Equations, 231 (2006), 32. doi: 10.1016/j.jde.2006.04.006. Google Scholar [3] G. Akagi and K. Matsuura, Nonlinear diffusion equations driven by the $p(\cdot)$-Laplacian,, Nonlinear Differential Equations and Applications (NoDEA), 20 (2013), 37. doi: 10.1007/s00030-012-0153-6. Google Scholar [4] G. Akagi, Energy solutions of the Cauchy-Neumann problem for porous medium equations,, Discrete and Continuous Dynamical Systems, (2009), 1. Google Scholar [5] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations,, Math. Z., 183 (1983), 311. doi: 10.1007/BF01176474. Google Scholar [6] S. Antontsev and S. Shmarev, Parabolic equations with double variable nonlinearities,, Math. Comput. Simulation, 81 (2011), 2018. doi: 10.1016/j.matcom.2010.12.015. Google Scholar [7] S. Antontsev and S. Shmarev, Extinction of solutions of parabolic equations with variable anisotropic nonlinearities,, Proc. Steklov Inst. Math., 261 (2008), 11. doi: 10.1134/S0081543808020028. Google Scholar [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. doi: 10.1016/j.na.2004.09.026. Google Scholar [9] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'', Noordhoff, (1976). Google Scholar [10] V. Barbu, Existence for nonlinear Volterra equations in Hilbert spaces,, SIAM J. Math. Anal., 10 (1979), 552. doi: 10.1137/0510052. Google Scholar [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. doi: 10.1016/j.jde.2010.05.011. Google Scholar [12] T. M. Bokalo and O. M. Buhrii, Doubly nonlinear parabolic equations with variable exponents of nonlinearity,, Ukrainian Math. J., 63 (2011), 709. doi: 10.1007/s11253-011-0537-5. Google Scholar [13] H. Brézis, "Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,'', Math Studies, (1973). Google Scholar [14] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration,, SIAM J. Appl. Math., 66 (2006), 1383. doi: 10.1137/050624522. Google Scholar [15] E. DiBenedetto and R. E. Showalter, Implicit degenerate evolution equations and applications,, SIAM J. Math. Anal., 12 (1981), 731. doi: 10.1137/0512062. Google Scholar [16] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, "Lebesgue and Sobolev Spaces with Variable Exponents,'', Lecture Notes in Mathematics, (2017). doi: 10.1007/978-3-642-18363-8. Google Scholar [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. doi: 10.1006/jmaa.2000.7617. Google Scholar [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. doi: 10.1006/jmaa.2000.7266. Google Scholar [19] Y. Fu and N. Pan, Existence of solutions for nonlinear parabolic problem with $p(x)$-growth,, J. Math. Anal. Appl., 362 (2010), 313. doi: 10.1016/j.jmaa.2009.08.038. Google Scholar [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. doi: 10.1016/0022-1236(72)90080-8. Google Scholar [21] P. Harjulehto, P. Hästö, Ú.-V. Lê and M. Nuortio, Overview of differential equations with non-standard growth,, Nonlinear Anal., 72 (2010), 4551. doi: 10.1016/j.na.2010.02.033. Google Scholar [22] N. Kenmochi, Some nonlinear parabolic variational inequalities,, Israel J. Math., 22 (1975), 304. doi: 10.1007/BF02761596. Google Scholar [23] N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with time-dependent constraints,, Nonlinear Anal., 10 (1986), 1181. doi: 10.1016/0362-546X(86)90058-1. Google Scholar [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. doi: 10.1016/j.jmaa.2007.07.083. Google Scholar [25] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires,'', Dunod, (1969). Google Scholar [26] J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications I,'', Die Grundlehren der mathematischen Wissenschaften, (1972). Google Scholar [27] E. Maitre and P. Witomski, A pseudo-monotonicity adapted to doubly nonlinear elliptic-parabolic equations,, Nonlinear Anal., 50 (2002), 223. doi: 10.1016/S0362-546X(01)00748-9. Google Scholar [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. doi: 10.5186/aasfm.2010.3507. Google Scholar [29] J. Musielak, "Orlicz Spaces and Modular Spaces,'', Lecture Notes in Mathematics, (1034). Google Scholar [30] P. A. Raviart, Sur la résolution de certaines équations paraboliques non linéaires,, J. Functional Analysis, 5 (1970), 299. doi: 10.1016/0022-1236(70)90031-5. Google Scholar [31] T. Roubíček, "Nonlinear Partial Differential Equations with Applications,'', International Series of Numerical Mathematics, (2005). Google Scholar [32] M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory,'' Lecture Notes in Mathematics, Vol. 1748,, Springer-Verlag, (2000). doi: 10.1007/BFb0104029. Google Scholar [33] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura. Appl. (4), 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [34] M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption,, J. Math. Anal. Appl., 132 (1988), 187. doi: 10.1016/0022-247X(88)90053-4. Google Scholar [35] N. Yamazaki, Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems,, Discrete and Continuous Dynamical Systems, (2005), 920. Google Scholar [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. doi: 10.1016/j.jde.2009.11.024. Google Scholar [37] V. V. Zhikov, On the technique for passing to the limit in nonlinear elliptic equations,, Functional Analysis and Its Applications, 43 (2009), 96. doi: 10.1007/s10688-009-0014-1. Google Scholar

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##### 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. doi: 10.1007/s002050100117. Google Scholar [2] G. Akagi, Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces,, J. Differential Equations, 231 (2006), 32. doi: 10.1016/j.jde.2006.04.006. Google Scholar [3] G. Akagi and K. Matsuura, Nonlinear diffusion equations driven by the $p(\cdot)$-Laplacian,, Nonlinear Differential Equations and Applications (NoDEA), 20 (2013), 37. doi: 10.1007/s00030-012-0153-6. Google Scholar [4] G. Akagi, Energy solutions of the Cauchy-Neumann problem for porous medium equations,, Discrete and Continuous Dynamical Systems, (2009), 1. Google Scholar [5] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations,, Math. Z., 183 (1983), 311. doi: 10.1007/BF01176474. Google Scholar [6] S. Antontsev and S. Shmarev, Parabolic equations with double variable nonlinearities,, Math. Comput. Simulation, 81 (2011), 2018. doi: 10.1016/j.matcom.2010.12.015. Google Scholar [7] S. Antontsev and S. Shmarev, Extinction of solutions of parabolic equations with variable anisotropic nonlinearities,, Proc. Steklov Inst. Math., 261 (2008), 11. doi: 10.1134/S0081543808020028. Google Scholar [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. doi: 10.1016/j.na.2004.09.026. Google Scholar [9] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'', Noordhoff, (1976). Google Scholar [10] V. Barbu, Existence for nonlinear Volterra equations in Hilbert spaces,, SIAM J. Math. Anal., 10 (1979), 552. doi: 10.1137/0510052. Google Scholar [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. doi: 10.1016/j.jde.2010.05.011. Google Scholar [12] T. M. Bokalo and O. M. Buhrii, Doubly nonlinear parabolic equations with variable exponents of nonlinearity,, Ukrainian Math. J., 63 (2011), 709. doi: 10.1007/s11253-011-0537-5. Google Scholar [13] H. Brézis, "Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,'', Math Studies, (1973). Google Scholar [14] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration,, SIAM J. Appl. Math., 66 (2006), 1383. doi: 10.1137/050624522. Google Scholar [15] E. DiBenedetto and R. E. Showalter, Implicit degenerate evolution equations and applications,, SIAM J. Math. Anal., 12 (1981), 731. doi: 10.1137/0512062. Google Scholar [16] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, "Lebesgue and Sobolev Spaces with Variable Exponents,'', Lecture Notes in Mathematics, (2017). doi: 10.1007/978-3-642-18363-8. Google Scholar [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. doi: 10.1006/jmaa.2000.7617. Google Scholar [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. doi: 10.1006/jmaa.2000.7266. Google Scholar [19] Y. Fu and N. Pan, Existence of solutions for nonlinear parabolic problem with $p(x)$-growth,, J. Math. Anal. Appl., 362 (2010), 313. doi: 10.1016/j.jmaa.2009.08.038. Google Scholar [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. doi: 10.1016/0022-1236(72)90080-8. Google Scholar [21] P. Harjulehto, P. Hästö, Ú.-V. Lê and M. Nuortio, Overview of differential equations with non-standard growth,, Nonlinear Anal., 72 (2010), 4551. doi: 10.1016/j.na.2010.02.033. Google Scholar [22] N. Kenmochi, Some nonlinear parabolic variational inequalities,, Israel J. Math., 22 (1975), 304. doi: 10.1007/BF02761596. Google Scholar [23] N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with time-dependent constraints,, Nonlinear Anal., 10 (1986), 1181. doi: 10.1016/0362-546X(86)90058-1. Google Scholar [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. doi: 10.1016/j.jmaa.2007.07.083. Google Scholar [25] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires,'', Dunod, (1969). Google Scholar [26] J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications I,'', Die Grundlehren der mathematischen Wissenschaften, (1972). Google Scholar [27] E. Maitre and P. Witomski, A pseudo-monotonicity adapted to doubly nonlinear elliptic-parabolic equations,, Nonlinear Anal., 50 (2002), 223. doi: 10.1016/S0362-546X(01)00748-9. Google Scholar [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. doi: 10.5186/aasfm.2010.3507. Google Scholar [29] J. Musielak, "Orlicz Spaces and Modular Spaces,'', Lecture Notes in Mathematics, (1034). Google Scholar [30] P. A. Raviart, Sur la résolution de certaines équations paraboliques non linéaires,, J. Functional Analysis, 5 (1970), 299. doi: 10.1016/0022-1236(70)90031-5. Google Scholar [31] T. Roubíček, "Nonlinear Partial Differential Equations with Applications,'', International Series of Numerical Mathematics, (2005). Google Scholar [32] M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory,'' Lecture Notes in Mathematics, Vol. 1748,, Springer-Verlag, (2000). doi: 10.1007/BFb0104029. Google Scholar [33] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura. Appl. (4), 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [34] M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption,, J. Math. Anal. Appl., 132 (1988), 187. doi: 10.1016/0022-247X(88)90053-4. Google Scholar [35] N. Yamazaki, Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems,, Discrete and Continuous Dynamical Systems, (2005), 920. Google Scholar [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. doi: 10.1016/j.jde.2009.11.024. Google Scholar [37] V. V. Zhikov, On the technique for passing to the limit in nonlinear elliptic equations,, Functional Analysis and Its Applications, 43 (2009), 96. doi: 10.1007/s10688-009-0014-1. Google Scholar
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