On the continuity of solutions of the equations of a porous medium and the fast diffusion with weighted and singular lower-order terms

2021 ◽  
Vol 18 (1) ◽  
pp. 104-139
Author(s):  
Yevhen Zozulia
Keyword(s):  
Porous Medium ◽  
Parabolic Equation ◽  
Harnack Inequality ◽  
Lower Order ◽  
Riesz Potential ◽  
Fast Diffusion ◽  
Right Hand ◽  
Continuity Of Solutions ◽  
The Right ◽  
Lower Order Terms

For the parabolic equation $$ \ v\left(x \right)u_{t} -{div({\omega(x)u^{m-1}}} \nabla u) = f(x,t)\: ,\; u\geq{0}\:,\; m\neq{1} $$ we prove the continuity and the Harnack inequality for generalized k solutions, by using the weighted Riesz potential on the right-hand side of the equation.

scholarly journals Hardy potential versus lower order terms in Dirichlet problems: regularizing effects

2022 ◽  
Vol 5 (1) ◽  
pp. 1-14
Author(s):  
David Arcoya ◽  
◽  
Lucio Boccardo ◽  
Luigi Orsina ◽  
◽  
...  
Keyword(s):  
Lower Order ◽  
Potential Versus ◽  
Dirichlet Problems ◽  
Hardy Potential ◽  
Right Hand ◽  
The Right ◽  
Regularizing Effects ◽  
Lower Order Terms

<abstract><p>In this paper, dedicated to Ireneo Peral, we study the regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy potentials in the right hand side.</p></abstract>

Pointwise estimates of solutions to the weighted porous medium equation and the fast diffusion one via weighted Riesz potentials

2020 ◽  
Vol 17 (1) ◽  
pp. 116-143
Author(s):  
Yevhen Zozulia
Keyword(s):  
Porous Medium ◽  
Riesz Potential ◽  
Porous Medium Equation ◽  
Fast Diffusion ◽  
Pointwise Estimates ◽  
Riesz Potentials ◽  
Local Boundedness ◽  
Estimates Of Solutions ◽  
Medium Equation ◽  
The Right

For the weighted parabolic equation $$ \ v\left(x \right)u_{t} - {div({\omega(x)u^{m-1}}} \nabla u) = f(x,t)\: ,\; u\geq{0}\:,\; m\neq{1}, $$ we prove the local boundedness of weak solutions in terms of the ${\;}$ weighted Riesz potential on the right-hand side of the equation.

scholarly journals Existence of Solutions for Some Nonlinear Elliptic Anisotropic Unilateral Problems with Lower Order Terms

2018 ◽  
Vol 4 (2) ◽  
pp. 171-188 ◽  
Author(s):  
Youssef Akdim ◽  
Chakir Allalou ◽  
Abdelhafid Salmani
Keyword(s):  
Lower Order ◽  
Existence Of Solutions ◽  
Entropy Solutions ◽  
Unilateral Problem ◽  
Nonlinear Elliptic ◽  
Right Hand ◽  
Unilateral Problems ◽  
The Right ◽  
Lower Order Terms

AbstractIn this paper, we prove the existence of entropy solutions for anisotropic elliptic unilateral problem associated to the equations of the form$$ - \sum\limits_{i = 1}^N {{\partial _i}{a_i}(x,u,\nabla u) - } \sum\limits_{i = 1}^N {{\partial _i}{\phi _i}(u) = f,} $$where the right hand side f belongs to L1(Ω). The operator $- \sum\nolimits_{i = 1}^N {{\partial _i}{a_i}\left( {x,u,\nabla u} \right)} $ is a Leray-Lions anisotropic operator and ϕi ∈ C0(ℝ,ℝ).

On the determination of the right-hand side in a parabolic equation

2012 ◽  
Vol 62 (11) ◽  
pp. 1672-1683 ◽  
Author(s):  
A. Ashyralyev ◽  
A.S. Erdogan ◽  
O. Demirdag
Keyword(s):  
Parabolic Equation ◽  
Right Hand ◽  
The Right

scholarly journals On the Harnack inequality for degenerate and singular elliptic equations with unbounded lower order terms via sliding paraboloids

2017 ◽  
Vol 20 (01) ◽  
pp. 1750012 ◽  
Author(s):  
Nam Q. Le
Keyword(s):  
Elliptic Equation ◽  
Harnack Inequality ◽  
Lower Order ◽  
Elliptic Equations ◽  
Singular Elliptic Equation ◽  
Large Gradient ◽  
Singular Elliptic Equations ◽  
Monge Ampere ◽  
Lower Order Terms ◽  
Linear Degenerate

We use the method of sliding paraboloids to establish a Harnack inequality for linear, degenerate and singular elliptic equation with unbounded lower order terms. The equations we consider include uniformly elliptic equations and linearized Monge–Ampère equations. Our argument allows us to prove the doubling estimate for functions which, at points of large gradient, are solutions of (degenerate and singular) elliptic equations with unbounded drift.

Sign in / Sign up

Export Citation Format

Share Document