Null controllability for a parabolic equation with nonlocal nonlinearities

2012 ◽  
Vol 61 (1) ◽  
pp. 107-111 ◽  
Author(s):  
Enrique Fernández-Cara ◽  
Juan Limaco ◽  
Silvano B. de Menezes
Keyword(s):  
Parabolic Equation ◽  
Null Controllability ◽  
Nonlocal Nonlinearities

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.

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