Nonlinear diffusion equations driven by the \(p(\cdot)\)-Laplacian
type: Item
instance of: scholarly article
title: Nonlinear diffusion equations driven by the \(p(\cdot)\)-Laplacian
zbMATH Open document ID: 1264.35136
author: Goro Akagi
author: Kei Matsuura
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2013-02-21T00:00:00Z
review text: The authors considered the following initial-boundary value problem: \[ \partial_tu=\nabla\cdot(|\nabla\phi(x)|^{p(x)-2}\nabla\phi(x))+f,\qquad\text{in }\Omega\times (0,\infty), \] \[ u=0,\qquad\text{on }\partial\Omega\times (0,\infty), \] \[ u(\cdot,0)=u_0,\qquad\text{in }\Omega, \] where \(\Omega\subset{\mathbb R}^N\) is a bounded domain with smooth boundary \(\partial\Omega\), \(f\) and \(u_0\) are given functions. The authors investigated the asymptotic behaviors of the solutions of the problem as \(t\rightarrow 0\) when \(f\equiv0\), and also found out the limiting behavior of the solutions as \(p_n(\cdot)\rightarrow\infty\).
reviewed by: Yuanyuan Ke
Mathematics Subject Classification ID: 35K92
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 6138201
zbMATH Keywords: subdifferential
zbMATH Keywords: variable exponent Lebesgue spaces
MaRDI profile type: Publication
full work available at URL: s00030-012-0153-6
OpenAlex ID: W1991179240
cites work: Regularity results for a class of functionals with non-standard growth
cites work: Regularity results for stationary electro-rheological fluids
cites work: Regularity results for parabolic systems related to a class of non-Newtonian fluids.
cites work: Convergence of functionals and its applications to parabolic equations
cites work: Well-posedness and large-time behaviors of solutions for a parabolic equation involving \(p(x)\)-Laplacian
cites work: Q5694396
cites work: Extinction of solutions of parabolic equations with variable anisotropic nonlinearities
cites work: Anisotropic parabolic equations with variable nonlinearity
cites work: Vanishing solutions of anisotropic parabolic equations with variable nonlinearity
cites work: Blow-up of solutions to parabolic equations with nonstandard growth conditions
cites work: Fast/slow diffusion and growing sandpiles
cites work: Q5186236
cites work: Bean's critical-state model as the \(p\rightarrow\infty\) limit of an evolutionary \(p\)-Laplacian equation
cites work: Stability of the separable solution for fast diffusion
cites work: Renormalized solutions for a nonlinear parabolic equation with variable exponents and \(L^{1}\)-data
cites work: Q5665783
cites work: Variable Exponent, Linear Growth Functionals in Image Restoration
cites work: Degenerate parabolic equations
cites work: \(C^{1,\alpha}\)-regularity for electrorheological fluids in two dimensions
cites work: Lebesgue and Sobolev spaces with variable exponents
cites work: Existence of local strong solutions for motions of electrorheological fluids in three dimensions
cites work: Sobolev embeddings with variable exponent
cites work: Q4797974
cites work: Sobolev embedding theorems for spaces \(W^{k,p(x)}(\Omega)\)
cites work: On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m,p(x)}(\Omega)\)
cites work: Compact imbedding theorems with symmetry of Strauss-Lions type for the space \(W^{1,p(x)}(\Omega)\)
cites work: Existence of solutions for nonlinear parabolic problem with \(p(x)\)-growth
cites work: Overview of differential equations with non-standard growth
cites work: Q4021390
cites work: \(p(x)\)-harmonic functions with unbounded exponent in a subdomain
cites work: Limits as \(p(x)\rightarrow \infty \) of \(p(x)\)-harmonic functions
cites work: Orlicz spaces and modular spaces
cites work: The Degeneracy of a Fast-Diffusion Equation and Stability
cites work: Electrorheological fluids: modeling and mathematical theory
cites work: On a progress in the theory of lebesgue spaces with variable exponent: maximal and singular operators
cites work: Entropy solutions for the $p(x)$-Laplace equation
cites work: On the well-posedness of a two-phase minimization problem
cites work: Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and \(L^{1}\) data
cites work: On the technique for passing to the limit in nonlinear elliptic equations
DOI: 10.1007/S00030-012-0153-6
Recommended article: Non-local diffusion equations involving the fractional \(p(\cdot)\)-Laplacian
Recommended article: Existence,uniqueness, and nonexistence of solutions to nonlinear diffusion equations with \(p(x,t)\)-Laplacian operator
Recommended article: A Nonlinear Pseudoparabolic Diffusion Equation
Recommended article: Q3158677
Recommended article: Q3222604
Recommended article: The uniqueness of a nonlinear diffusion equation related to the \(p\)-Laplacian
Recommended article: On the nonlinear diffusion equation of Kolmogorov, Petrovskij, and Piscunov
Recommended article: Q5416026
Recommended article: Non-self-similar solutions of multidimensional nonlinear diffusion equations
Recommended article: Nonlinear Diffusion with Fractional Laplacian Operators
description: scientific article; zbMATH DE number 6138201