scholarly journals On a general degenerate/singular parabolic equation with a nonlocal space term

2021 ◽  
Author(s):  
Brahim Allal ◽  
Genni Fragnelli ◽  
Jawad Salhi
Keyword(s):  
Fixed Point ◽  
Parabolic Equation ◽  
Null Controllability ◽  
Carleman Estimates ◽  
Heat Equations ◽  
Well Posedness ◽  
Nonlocal Problems ◽  
Singular Parabolic Equation ◽  
Controllability Property ◽  
T Tau

In this paper we study the null controllability for the problems associated to the operators y_t-Ay - \lambda/b(x) y+\int_0^1 K(t,x,\tau)y(t, \tau) d\tau, (t,x) \in (0,T)\times (0,1) where Ay := ay_{xx} or Ay := (ay_x)_x and the functions a and b degenerate at an interior point x0 Ë .0; 1/. To this aim, as a first step we study the well posedness, the Carleman estimates and the null controllability for the associated nonhomogeneous degenerate and singular heat equations. Then,using the Kakutani’s fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.

Controllability Property of a Superlinear Climate System

2013 ◽  
Vol 367 ◽  
pp. 264-269
Author(s):  
Liang Zhang ◽  
Yang Liu
Keyword(s):  
Fixed Point ◽  
Parabolic Equation ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Point Of View ◽  
Climate System ◽  
Linear Parabolic Equation ◽  
Controllability Property

This work concerns a climate system in the point of view of controllability. We obtain by the Kakutani’s fixed point theorem and the controllability property of the linear parabolic equation that the superlinear climate system is null controllable in the case with interior control.

scholarly journals NULL CONTROLLABILITY OF DEGENERATE NONAUTONOMOUS PARABOLIC EQUATIONS

2019 ◽  
pp. 311
Author(s):  
Abbes Benaissa ◽  
Abdelatif Kainane Mezadek ◽  
Lahcen Maniar
Keyword(s):  
Parabolic Equation ◽  
Open Subset ◽  
Parabolic Equations ◽  
Diffusion Coefficients ◽  
Adjoint System ◽  
Null Controllability ◽  
Carleman Estimates ◽  
One Dimensional ◽  
Neumann Type ◽  
The One

In this paper we are interested in the study of the null controllability for the one dimensional degenerate non autonomous parabolic equation$$u_{t}-M(t)(a(x)u_{x})_{x}=h\chi_{\omega},\qquad  (x,t)\in Q=(0,1)\times(0,T),$$ where $\omega=(x_{1},x_{2})$ is asmall nonempty open subset in $(0,1)$, $h\in L^{2}(\omega\times(0,T))$, the diffusion coefficients $a(\cdot)$ isdegenerate at $x=0$ and $M(\cdot)$ is non degenerate on $[0,T]$. Also the boundary conditions are considered tobe Dirichlet or Neumann type related to the degeneracy rate of $a(\cdot)$. Under some conditions on the functions$a(\cdot)$ and $M(\cdot)$, we prove some global Carleman estimates which will yield the  observability inequalityof the associated adjoint system and equivalently the null controllability of our parabolic equation.

scholarly journals Exact Null Controllability of KdV-Burgers Equation with Memory Effect Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Rajagounder Ravi Kumar ◽  
Kil To Chong ◽  
Jong Ho Park
Keyword(s):  
Burgers Equation ◽  
Null Controllability ◽  
Carleman Estimates ◽  
External Control ◽  
Observability Inequality ◽  
Exact Null Controllability ◽  
Controllability Property ◽  
The Stability ◽  
With Memory

This paper is concerned with exact null controllability analysis of nonlinear KdV-Burgers equation with memory. The proposed approach relies upon regression tool to prove controllability property of linearized KdV-Burgers equation via Carleman estimates. The control is distributed along with subdomainω⊂Ωand the external control acts on the key role of observability inequality with memory. This description finally showed the exact null controllability guaranteeing the stability.

scholarly journals The BV solution of the parabolic equation with degeneracy on the boundary

2016 ◽  
Vol 14 (1) ◽  
pp. 272-282
Author(s):  
Huashui Zhan ◽  
Shuping Chen
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Well Posedness ◽  
Test Functions ◽  
The Stability

AbstractConsider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

Inverse problem for a degenerate/singular parabolic system with Neumann boundary conditions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammed Alaoui ◽  
Abdelkarim Hajjaj ◽  
Lahcen Maniar ◽  
Jawad Salhi
Keyword(s):  
Boundary Conditions ◽  
Parabolic System ◽  
Source Term ◽  
Neumann Boundary ◽  
Stability Estimate ◽  
Carleman Estimates ◽  
Well Posedness ◽  
Degenerate Diffusion ◽  
Neumann Type ◽  
Lower Order Terms

AbstractIn this paper, we study an inverse source problem for a degenerate and singular parabolic system where the boundary conditions are of Neumann type. We consider a problem with degenerate diffusion coefficients and singular lower-order terms, both vanishing at an interior point of the space domain. In particular, we address the question of well-posedness of the problem, and then we prove a stability estimate of Lipschitz type in determining the source term by data of only one component. Our method is based on Carleman estimates, cut-off procedures and a reflection technique.

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