scholarly journals Nonlinear evolution problems with singular coefficients in the lower order terms

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Fernando Farroni ◽  
Luigi Greco ◽  
Gioconda Moscariello ◽  
Gabriella Zecca
Keyword(s):  
Lower Order ◽  
Time Horizon ◽  
Existence Result ◽  
Nonlinear Evolution ◽  
Order Term ◽  
Time Behavior ◽  
Infinite Time ◽  
Singular Coefficient ◽  
Singular Coefficients ◽  
Lower Order Terms

AbstractWe consider a Cauchy–Dirichlet problem for a quasilinear second order parabolic equation with lower order term driven by a singular coefficient. We establish an existence result to such a problem and we describe the time behavior of the solution in the case of the infinite–time horizon.

scholarly journals Existence of Solutions to Elliptic Problem with Convection Term and Lower-Order Term

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Xiaohua He ◽  
Shuibo Huang ◽  
Qiaoyu Tian ◽  
Yonglin Xu
Keyword(s):  
Dirichlet Problem ◽  
Lower Order ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Order Term ◽  
Combined Effects ◽  
Convection Term ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Lower Order Terms

In this paper, we establish the existence of solutions to the following noncoercivity Dirichlet problem − div M x ∇ u + u p − 1 u = − div u E x + f x , x ∈ Ω , u x = 0 , x ∈ ∂ Ω , where Ω ⊂ ℝ N N > 2 is a bounded smooth domain with 0 ∈ Ω , f belongs to the Lebesgue space L m Ω with m ≥ 1 , p > 0 . The main innovation point of this paper is the combined effects of the convection terms and lower-order terms in elliptic equations.

scholarly journals A non-existence result for nonlinear parabolic equations with singular measures as data

2009 ◽  
Vol 139 (2) ◽  
pp. 381-392 ◽  
Author(s):  
Francesco Petitta
Keyword(s):  
Laplace Operator ◽  
Lower Order ◽  
Parabolic Equations ◽  
Existence Result ◽  
Nonlinear Parabolic Equations ◽  
Parabolic Problems ◽  
Singular Measures ◽  
Nonlinear Parabolic ◽  
Image Position ◽  
Lower Order Terms

In this paper we prove a non-existence result for nonlinear parabolic problems with zero lower-order terms whose model iswhere Δp=div(|∇u|p−2∇u) is the usual p-laplace operator, λ is measure concentrated on a set of zero parabolic r-capacity (1<p<r) and q is large enough.

EXISTENCE RESULT FOR NONLINEAR PARABOLIC EQUATIONS WITH LOWER ORDER TERMS

2011 ◽  
Vol 09 (02) ◽  
pp. 161-186 ◽  
Author(s):  
ROSARIA DI NARDO ◽  
FILOMENA FEO ◽  
OLIVIER GUIBÉ
Keyword(s):  
Laplace Operator ◽  
Lower Order ◽  
Parabolic Equations ◽  
Existence Result ◽  
Nonlinear Parabolic Equations ◽  
Parabolic Problems ◽  
Bounded Subset ◽  
Renormalized Solution ◽  
Nonlinear Parabolic ◽  
Lower Order Terms

In this paper, we prove, the existence of a renormalized solution for a class of nonlinear parabolic problems whose prototype is [Formula: see text] where QT = Ω × (0, T), Ω is an open and bounded subset of ℝN, N ≥ 2, T > 0, Δp is the so called p-Laplace operator, [Formula: see text], c ∈ (Lr(QT))N with [Formula: see text], [Formula: see text], b ∈ LN+2, 1(QT), f ∈ L1(QT), g ∈ (Lp'(QT))N and u0 ∈ L1(Ω).

Variational Inequalities with General Multivalued Lower Order Terms Given by Integrals

2011 ◽  
Vol 11 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Vy Khoi
Keyword(s):  
Variational Inequalities ◽  
Lower Order ◽  
Order Term ◽  
Multivalued Function ◽  
Solution Set ◽  
Lower Order Term ◽  
Extremal Solutions ◽  
Lower Order Terms

AbstractThis paper is about the existence and some properties of solutions of variational inequalities associated with the 2nd order inclusiondiv[A(x, ∇u)] + L ∈ f (x, u) in Ω,where the lower order term f (x, u) is a general multivalued function. Both coercive and noncoercive cases are considered. In the noncoercive case, we use a sub-supersolution approach to study the existence, comparison, and other properties of the solution set such as its compactness, directedness, and the existence of extremal solutions.

The Regularizing Effect of Lower Order Terms in Elliptic Problems Involving Hardy Potential

2017 ◽  
Vol 17 (2) ◽  
Author(s):  
A. Adimurthi ◽  
Lucio Boccardo ◽  
G. Rita Cirmi ◽  
Luigi Orsina
Keyword(s):  
Lower Order ◽  
Order Term ◽  
Elliptic Problems ◽  
Lower Order Term ◽  
Hardy Potential ◽  
Lower Order Terms

AbstractWe study existence and summability of solutions for elliptic problems with a power-like lower order term and a Hardy potential. We prove that, due to the presence of the lower order term, solutions exist and are more summable under weaker assumptions than those needed for the existence without it.

scholarly journals A regularity result for a class of elliptic equations with lower order terms

2020 ◽  
Vol 6 (2) ◽  
pp. 751-771 ◽  
Author(s):  
Claudia Capone ◽  
Teresa Radice
Keyword(s):  
Dirichlet Problem ◽  
Lower Order ◽  
Elliptic Equations ◽  
Order Term ◽  
Regularity Result ◽  
Lower Order Term ◽  
Bounded Solutions ◽  
Differentiability Of Solutions ◽  
Higher Differentiability ◽  
Lower Order Terms

Abstract In this paper we establish the higher differentiability of solutions to the Dirichlet problem $$\begin{aligned} {\left\{ \begin{array}{ll} \text {div} (A(x, Du)) + b(x)u(x)=f &{} \text {in}\, \Omega \\ u=0 &{} \text {on} \, \partial \Omega \end{array}\right. } \end{aligned}$$ div ( A ( x , D u ) ) + b ( x ) u ( x ) = f in Ω u = 0 on ∂ Ω under a Sobolev assumption on the partial map $$x \rightarrow A(x, \xi )$$ x → A ( x , ξ ) . The novelty here is that we take advantage from the regularizing effect of the lower order term to deal with bounded solutions.

scholarly journals Quasilinear degenerate elliptic unilateral problems

2005 ◽  
Vol 2005 (1) ◽  
pp. 11-31 ◽  
Author(s):  
L. Aharouch ◽  
Y. Akdim ◽  
E. Azroul
Keyword(s):  
Lower Order ◽  
Existence Result ◽  
Order Term ◽  
Natural Growth ◽  
Unilateral Problem ◽  
Sign Condition ◽  
Carathéodory Function ◽  
Degenerate Elliptic ◽  
Unilateral Problems ◽  
The Right

We will be concerned with the existence result of a degenerate elliptic unilateral problem of the formAu+H(x,u,∇u)=f, whereAis a Leray-Lions operator fromW1,p(Ω,w)into its dual. On the nonlinear lower-order termH(x,u,∇u), we assume that it is a Carathéodory function having natural growth with respect to|∇u|, but without assuming the sign condition. The right-hand sidefbelongs toL1(Ω).

scholarly journals Variational Inequalities with Multivalued Lower Order Terms and Convex Functionals in Orlicz-Sobolev Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Ge Dong ◽  
Xiaochun Fang
Keyword(s):  
Sobolev Spaces ◽  
Lower Order ◽  
Principal Part ◽  
Existence Of Solutions ◽  
Order Term ◽  
Convex Functional ◽  
Polynomial Type ◽  
Convex Functionals ◽  
Order Elliptic Operator ◽  
Lower Order Terms

We consider the existence of solutions of variational inequality form. Findu∈D(J):〈A(u),v-u〉+〈F(u),v-u〉+J(v)-J(u)≥0,∀v∈W1LM(Ω),whose principal part is having a growth not necessarily of polynomial type, whereAis a second-order elliptic operator of Leray-Lions type,Fis a multivalued lower order term, andJis a convex functional. We use subsupersolution methods to study the existence and enclosure of solutions in Orlicz-Sobolev spaces.

scholarly journals Renormalized solutions for nonlinear parabolic problems with L1−data in orlicz-sobolev spaces

2017 ◽  
Vol 35 (1) ◽  
pp. 57 ◽  
Author(s):  
Youssef El hadfi ◽  
Abdelmoujib Benkirane ◽  
Mostafa El moumni
Keyword(s):  
Sobolev Spaces ◽  
Lower Order ◽  
Parabolic Equations ◽  
Existence Result ◽  
Nonlinear Parabolic Equations ◽  
Parabolic Problems ◽  
Renormalized Solutions ◽  
L1 Data ◽  
Nonlinear Parabolic ◽  
Lower Order Terms

In this work, we prove an existence result of renormalized solutions in Orlicz-Sobolev spaces for a class of nonlinear parabolic equations with two lower order terms and L1-data. 

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