American Institute of Mathematical Sciences

Advanced
2011, 2011(Special): 22-31. doi: 10.3934/proc.2011.2011.22

Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian

Goro Akagi 1, and  Kei Matsuura 2,

1. 

Department of Machinery and Control Systems, College of Systems Engineering and Science,, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama-shi, Saitama 337-8570
2. 

Research Institute for Science Engineering, Waseda University, Okubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan

Received  July 2010 Revised  August 2011 Published  September 2011

This paper is concerned with the initial-boundary value problem for a nonlinear parabolic equation involving the so-called $p(x)$-Laplacian. A subdifferential approach is employed to obtain a well-posedness result as well as to investigate large-time behaviors of solutions.
Keywords: $p(x)$-Laplacian, subdifferential, parabolic equation, variable exponent Lebesgue and Sobolev spaces.
Mathematics Subject Classification: Primary: 35K55, 46E30; Secondary: 35K90.
Citation: 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
[1]

Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[2]

SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026
[3]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005
[4]

Pelin G. Geredeli, Azer Khanmamedov. Long-time dynamics of the parabolic $p$-Laplacian equation. Communications on Pure & Applied Analysis, 2013, 12 (2) : 735-754. doi: 10.3934/cpaa.2013.12.735
[5]

Shun Uchida. Solvability of doubly nonlinear parabolic equation with p-laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021033
[6]

Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[7]

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
[8]

Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51.

[9]

Valerii Los, Vladimir Mikhailets, Aleksandr Murach. Parabolic problems in generalized Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3589-3620. doi: 10.3934/cpaa.2021123
[10]

Anouar Bahrouni, VicenŢiu D. RĂdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 379-389. doi: 10.3934/dcdss.2018021
[11]

Ángel Arroyo, Joonas Heino, Mikko Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-laplacian. Communications on Pure & 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 & Continuous Dynamical Systems, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175
[13]

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 & Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731
[14]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231
[15]

Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053
[16]

Carla Baroncini, Julián Fernández Bonder. An extension of a Theorem of V. Šverák to variable exponent spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1987-2007. doi: 10.3934/cpaa.2015.14.1987
[17]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103
[18]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058
[19]

Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131
[20]

Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143

 Impact Factor: 

Tools

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences