Equivalence between viscosity and weak solutions for the parabolic equations with nonstandard growth

2021 ◽  
Author(s):  
Yuzhou Fang ◽  
Chao Zhang
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Nonstandard Growth

\mathfrak{B}_{1} classes of De Giorgi, Ladyzhenskaya, and Ural'tseva and their application to elliptic and parabolic equations with nonstandard growth

2019 ◽  
Vol 16 (3) ◽  
pp. 403-447
Author(s):  
Igor Skrypnik ◽  
Mykhailo Voitovych
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Hölder Continuity ◽  
Growth Conditions ◽  
Holder Continuity ◽  
Nonstandard Growth ◽  
Elliptic And Parabolic Equations ◽  
Functional Classes ◽  
Nonstandard Growth Conditions ◽  
Quasilinear Elliptic

The article provides an application of the generalized De Giorgi functional classes to the proof of the Hölder continuity of weak solutions to quasilinear elliptic and parabolic equations with nonstandard growth conditions.

scholarly journals Local subestimates of solutions to double-phase parabolic equations via nonlinear parabolic potentials

2019 ◽  
Vol 16 (1) ◽  
pp. 28-45
Author(s):  
Kateryna Buryachenko
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Growth Conditions ◽  
Local Boundedness ◽  
Nonstandard Growth ◽  
Right Hand ◽  
Double Phase ◽  
Nonlinear Parabolic ◽  
Nonstandard Growth Conditions ◽  
The Right

For parabolic equations with nonstandard growth conditions, we prove local boundedness of weak solutions in terms of nonlinear parabolic potentials of the right-hand side of the equation.

scholarly journals A Boundary Estimate for Singular Sub-Critical Parabolic Equations

2020 ◽  
Author(s):  
Ugo Gianazza ◽  
Naian Liao
Keyword(s):  
Modulus Of Continuity ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Boundary Point ◽  
Cylindrical Domain ◽  
Boundary Estimate ◽  
Critical Range ◽  
Type Integral ◽  
Wiener Type ◽  
Singular Parabolic Equations

Abstract We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of $p$-Laplacian type, with $p$ in the sub-critical range $\big(1,\frac{2N}{N+1}\big]$. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.

scholarly journals Hölder regularity for degenerate parabolic equations with variable exponents

2022 ◽  
Vol 40 ◽  
pp. 1-19
Author(s):  
Hamid EL Bahja
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equations ◽  
Hölder Regularity ◽  
Variable Exponents ◽  
Young’S Inequality ◽  
Scaling Method ◽  
Degenerate Parabolic ◽  
Young's Inequality ◽  
Holder Regularity

In this paper, we discuss a class of degenerate parabolic equations with variable exponents. By  using the Steklov average and Young's inequality, we establish energy and logarithmicestimates for solutions to these equations. Then based on the intrinsic scaling method, we provethat local weak solutions are locally continuous.

Weak Solutions for Parabolic Equations

2018 ◽  
pp. 343-364
Author(s):  
Marin Marin ◽  
Andreas Öchsner
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations
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