Quasilinear parabolic problem with p(x)-laplacian: existence, uniqueness of weak solutions and stabilization

Giacomoni, Jacques; Tiwari, Sweta; Warnault, Guillaume

doi:10.1007/s00030-016-0380-3

Cited by 23 publications

(14 citation statements)

References 25 publications

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“…Proof. A similar result exists in [13] or in [16] without the term k β |A k,R | p p * +ε . For the reader's convenience, we include the complete proof.…”

Section: An Application To Nonhomogeneous Operatorssupporting

confidence: 70%

“…So, it is relevant to establish a new version of the Picone identity to include a large class of nonstandard p(x)-growth problems. In [14,16,22] convexity arguments to homogeneous functionals have been used to deal with quasilinear elliptic and parabolic problems with variable exponents. In the present paper, taking advantage of our new Picone pointwise identity, we give further applications in the context of elliptic and parabolic problems.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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A Picone identity for variable exponent operators and applications

Arora¹,

Giacomoni²,

Warnault³

2019

Advances in Nonlinear Analysis

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In this work, we establish a new Picone identity for anisotropic quasilinear operators, such as the p(x)-Laplacian defined as div(|∇u| p(x)−2 ∇u). Our extension provides a new version of the Diaz-Saa inequality and new uniqueness results to some quasilinear elliptic equations with variable exponents. This new Picone identity can be also used to prove some accretivity property to a class of fast diffusion equations involving variable exponents. Using this, we prove for this class of parabolic equations a new weak comparison principle.

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“…Proof. A similar result exists in [13] or in [16] without the term k β |A k,R | p p * +ε . For the reader's convenience, we include the complete proof.…”

Section: An Application To Nonhomogeneous Operatorssupporting

confidence: 70%

Section: Introduction and Main Resultsmentioning

confidence: 99%

A Picone identity for variable exponent operators and applications

Arora¹,

Giacomoni²,

Warnault³

2019

Advances in Nonlinear Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…In specific, the importance of investigating these kinds of problems lies in modeling of various anisotropic features that occur in electrorheological flows, image restoration, filtration process in complex media, stratigraphy problems and heterogeneous biological interactions. In the literature, there are many works that explore the questions of existence (local or global), regularity or behaviour of solutions for parabolic equations with variable exponent, for example [1,2,3,6,10,29,30]. Prior investigations have implemented diverse approaches to study the elliptic and parabolic problems with nonstandard growth.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%