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

Goro Akagi; Kei Matsuura

doi:10.3934/proc.2011.2011.22

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2011, Volume 2011: 22-31. Doi: 10.3934/proc.2011.2011.22 open access

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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

Received: June 30, 2010

Revised: July 31, 2011

Published: August 2017

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Abstract

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.

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