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Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent

Charkaoui Abderrahim, Fahim Houda, Alaa Nour Eddine

Opuscula Mathematica

2021

Vol. 41, no. 1

25--53

We are concerned with the existence of solutions to a class of quasilinear parabolic equations having critical growth nonlinearity with respect to the gradient and variable exponent. Using Schaeffer’s fixed point theorem combined with the sub- and supersolution method, we prove the existence results of a weak solutions to the considered problems.

Blow-up of solutions for a Kirchhoff type equation with variable-exponent nonlinearities

Shahrouzi Mohammad, Kargarfard Firoozeh

Journal of Applied Analysis

2021

Vol. 27, nr 1

97--105

This paper deals with a Kirchhoff type equation with variable exponent nonlinearities, subject to a nonlinear boundary condition.Under appropriate conditions and regarding arbitrary positive initial energy, it is proved that solutions blow up in a finite time.Moreover, we obtain the upper bound estimate of the blowup time.

Isotropic and anisotropic double-phase problems: old and new

Radulescu Vicentiu D.

Opuscula Mathematica

2019

Vol. 39, no. 2

259--279

We are concerned with the study of two classes of nonlinear problems driven by differential operators with unbalanced growth, which generalize the (p, q)- and (p(x),q(x))-Laplace operators. The associated energy is a double-phase functional, either isotropic or anisotropic. The content of this paper is in relationship with pioneering contributions due to P. Marcelimi and G. Mingione.

A Natural Class of Sequential Banach Spaces

Talponen J.

Bulletin of the Polish Academy of Sciences. Mathematics

2011

Vol. 59, no 2

185--196

We introduce and study a natural class of variable exponent ℓp spaces, which generalizes the classical spaces ℓp and c0. These spaces will typically not be rearrangement-invariant but instead they enjoy a good local control of some geometric properties. Some geometric examples are constructed by using these spaces.