-
Previous Article
A thermohydraulics model with temperature dependent constraint on velocity fields
- DCDS-S Home
- This Issue
-
Next Article
Analysis of non-equilibrium evolution problems: Selected topics in material and life sciences
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 |
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. |
[1] |
Shun Uchida. Solvability of doubly nonlinear parabolic equation with p-laplacian. Evolution Equations and Control Theory, 2022, 11 (3) : 975-1000. doi: 10.3934/eect.2021033 |
[2] |
Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22 |
[3] |
Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51 |
[4] |
Kerstin Does. An evolution equation involving the normalized $P$-Laplacian. Communications on Pure and Applied Analysis, 2011, 10 (1) : 361-396. doi: 10.3934/cpaa.2011.10.361 |
[5] |
Vitali Liskevich, Igor I. Skrypnik, Zeev Sobol. Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731 |
[6] |
Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920 |
[7] |
Pelin G. Geredeli, Azer Khanmamedov. Long-time dynamics of the parabolic $p$-Laplacian equation. Communications on Pure and Applied Analysis, 2013, 12 (2) : 735-754. doi: 10.3934/cpaa.2013.12.735 |
[8] |
Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations and Control Theory, 2022, 11 (2) : 399-414. doi: 10.3934/eect.2021005 |
[9] |
Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure and Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 |
[10] |
Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361 |
[11] |
Ángel Arroyo, Joonas Heino, Mikko Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-laplacian. Communications on Pure and Applied Analysis, 2017, 16 (3) : 915-944. doi: 10.3934/cpaa.2017044 |
[12] |
Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175 |
[13] |
Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure and Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475 |
[14] |
Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090 |
[15] |
Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 |
[16] |
Michinori Ishiwata. Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent. Conference Publications, 2005, 2005 (Special) : 443-452. doi: 10.3934/proc.2005.2005.443 |
[17] |
Antonio Segatti. Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 801-820. doi: 10.3934/dcds.2006.14.801 |
[18] |
Gabriele Bonanno, Giuseppina D'Aguì, Angela Sciammetta. One-dimensional nonlinear boundary value problems with variable exponent. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 179-191. doi: 10.3934/dcdss.2018011 |
[19] |
Huiling Li, Xiaoliu Wang, Xueyan Lu. A nonlinear Stefan problem with variable exponent and different moving parameters. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1671-1698. doi: 10.3934/dcdsb.2019246 |
[20] |
Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]