scholarly journals Fractional $$({\varvec{s}},{\varvec{p}})$$-Robin–Venttsel’ problems on extension domains

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Simone Creo ◽  
Maria Rosaria Lancia
Keyword(s):  
Weak Solution ◽  
Strong Solution ◽  
Elliptic Problem ◽  
Existence And Uniqueness ◽  
Type Problem ◽  
Extension Domain ◽  
Semigroup Approach ◽  
Global Boundedness

AbstractWe study a nonlocal Robin–Venttsel’-type problem for the regional fractional p-Laplacian in an extension domain $$\Omega $$ Ω with boundary a d-set. We prove existence and uniqueness of a strong solution via a semigroup approach. Markovianity and ultracontractivity properties are proved. We then consider the elliptic problem. We prove existence, uniqueness and global boundedness of the weak solution.

scholarly journals Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation

2020 ◽  
Vol 18 (1) ◽  
pp. 1302-1316
Author(s):  
Guobing Fan ◽  
Zhifeng Yang
Keyword(s):  
Boundary Condition ◽  
Optimal Control ◽  
Weak Solution ◽  
Existence And Uniqueness ◽  
Optimal Solution ◽  
Dispersive Equation ◽  
Weak Dissipation ◽  
Formula Type

Abstract In this paper, we investigate the problem for optimal control of a viscous generalized \theta -type dispersive equation (VG \theta -type DE) with weak dissipation. First, we prove the existence and uniqueness of weak solution to the equation. Then, we present the optimal control of a VG \theta -type DE with weak dissipation under boundary condition and prove the existence of optimal solution to the problem.

A piezoelectric contact problem with slip dependent friction and damage

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Abderrezak Kasri
Keyword(s):  
Weak Solution ◽  
Contact Problem ◽  
Dry Friction ◽  
Existence And Uniqueness ◽  
Variational Formulation ◽  
Normal Compliance ◽  
Electrically Conductive ◽  
Coulomb’S Law ◽  
Electrical Condition ◽  
Coulomb's Law

Abstract The aim of this paper is to study a quasistatic contact problem between an electro-elastic viscoplastic body with damage and an electrically conductive foundation. The contact is modelled with an electrical condition, normal compliance and the associated version of Coulomb’s law of dry friction in which slip dependent friction is included. We derive a variational formulation for the model and, under a smallness assumption, we prove the existence and uniqueness of a weak solution.

Existence and regularity of time-dependent pullback attractors for the non-autonomous nonclassical diffusion equations

2021 ◽  
pp. 1-18
Author(s):  
Yuming Qin ◽  
Bin Yang
Keyword(s):  
Weak Solution ◽  
Existence And Uniqueness ◽  
Nonlinear Term ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Diffusion ◽  
Force Term ◽  
Pullback Attractors ◽  
Critical Exponential Growth ◽  
Nonclassical Diffusion Equations

In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term $h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$ Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space $\mathcal {H}_{t}(\Omega )$ and the existence and regularity of the pullback attractors.

scholarly journals Nonstandard Dirichlet problems with competing (p,q)-Laplacian, convection, and convolution

2021 ◽  
Vol 66 (1) ◽  
pp. 95-103
Author(s):  
Dumitru Motreanu ◽  
Viorica Venera Motreanu
Keyword(s):  
Fixed Point ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Elliptic Problem ◽  
Integrable Function ◽  
Generalized Solution ◽  
Dirichlet Problems ◽  
Reaction Term ◽  
Fixed Point Approach

"The paper focuses on a nonstandard Dirichlet problem driven by the operator $-\Delta_p +\mu\Delta_q$, which is a competing $(p,q)$-Laplacian with lack of ellipticity if $\mu>0$, and exhibiting a reaction term in the form of a convection (i.e., it depends on the solution and its gradient) composed with the convolution of the solution with an integrable function. We prove the existence of a generalized solution through a combination of fixed-point approach and approximation. In the case $\mu\leq 0$, we obtain the existence of a weak solution to the respective elliptic problem."

scholarly journals On a Nonlinear Wave Equation of Kirchhoff-Carrier Type: Linear Approximation and Asymptotic Expansion of Solution in a Small Parameter

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Nguyen Huu Nhan ◽  
Le Thi Phuong Ngoc ◽  
Nguyen Thanh Long
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Small Parameter ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Existence And Uniqueness ◽  
Linearization Method ◽  
Asymptotic Expansion Of Solution

We consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed.

Existence and Uniqueness of a Weak Solution to a Stratigraphic Model

2004 ◽  
pp. 278-287
Author(s):  
Robert Eymard ◽  
Thierry Gallouët ◽  
Véronique Gervais ◽  
Roland Masson
Keyword(s):  
Weak Solution ◽  
Existence And Uniqueness ◽  
Stratigraphic Model

ON THE FIRST BOUNDARY VALUE PROBLEM FOR THE CLASS OF ELLIPTIC SYSTEMS DEGENERATING AT AN INNER POINT

2001 ◽  
Vol 6 (1) ◽  
pp. 147-155 ◽  
Author(s):  
S. Rutkauskas
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Elliptic Systems ◽  
Second Order ◽  
Type Problem ◽  
Dirichlet Type ◽  
Holder Class ◽  
Uniqueness Of The Solution ◽  
Dirichlet Type Problem

The Dirichlet type problem for the weakly related elliptic systems of the second order degenerating at an inner point is discussed. Existence and uniqueness of the solution in the Holder class of the vector‐functions is proved.

scholarly journals Analysis of a non-linear model of populations structured by size

2020 ◽  
Vol 101 (3) ◽  
pp. 734-750
Author(s):  
Anna Lo Grasso ◽  
Silvia Totaro
Keyword(s):  
Linear Model ◽  
Strong Solution ◽  
Mild Solution ◽  
Existence And Uniqueness ◽  
Marine Invertebrates ◽  
Life Stage ◽  
Pelagic Larvae ◽  
Non Linear ◽  
Local Habitat

AbstractThe model we study deals with a population of marine invertebrates structured by size whose life stage is composed of adults and pelagic larvae such as barnacles contained in a local habitat. We prove existence and uniqueness of a continuous positive global mild solution and we give an estimate of it. We prove also that this solution is the strong solution of the problem.

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