Doubly nonlinear parabolic equations       involving variable exponents

Goro Akagi

doi:10.3934/dcdss.2014.7.1

Article Contents

2014, Volume 7, Issue 1: 1-16. Doi: 10.3934/dcdss.2014.7.1

This issue Previous Article Analysis of non-equilibrium evolution problems: Selected topics in material and life sciences Next Article A thermohydraulics model with temperature dependent constraint on velocity fields

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

Received: January 31, 2012

Revised: December 31, 2012

Published: January 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K92, 47J35; Secondary: 46E30.

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