scholarly journals Carleman estimates for time-discrete parabolic equations and applications to controllability

2020 ◽  
Vol 26 ◽  
pp. 12
Author(s):  
Franck Boyer ◽  
Víctor Hernández-Santamaría
Keyword(s):  
Parabolic Equations ◽  
Linear Time ◽  
Parabolic Systems ◽  
Carleman Estimate ◽  
Carleman Estimates ◽  
Discretization Scheme ◽  
Parabolic Operator ◽  
Time Step ◽  
Coupled Parabolic Systems ◽  
Observability Estimates

In this paper, we prove a Carleman estimate for a time-discrete parabolic operator under some condition relating the large Carleman parameter to the time step of the discretization scheme. This estimate is then used to obtain relaxed observability estimates that yield, by duality, some controllability results for linear and semi-linear time-discrete parabolic equations. We also discuss the application of this Carleman estimate to the controllability of time-discrete coupled parabolic systems.

Stability in 𝐿𝑞-norm for inverse source parabolic problems

2020 ◽  
Vol 28 (6) ◽  
pp. 797-814
Author(s):  
Elena-Alexandra Melnig
Keyword(s):  
Parabolic Equations ◽  
Main Tool ◽  
Parabolic Systems ◽  
Carleman Estimates ◽  
Parabolic Problems ◽  
Lebesgue Spaces ◽  
First Order ◽  
Lipschitz Estimates ◽  
Systems Of Parabolic Equations

AbstractWe consider systems of parabolic equations coupled in zero and first order terms. We establish Lipschitz estimates in {L^{q}}-norms, {2\leq q\leq\infty}, for the source in terms of the solution in a subdomain. The main tool is a family of appropriate Carleman estimates with general weights, in Lebesgue spaces, for nonhomogeneous parabolic systems.

scholarly journals Global existence and nonexistence for a class of finitely degenerate coupled parabolic systems with high initial energy level

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yuxuan Chen ◽  
Jiangbo Han
Keyword(s):  
Energy Level ◽  
Global Existence ◽  
Blow Up ◽  
Parabolic Systems ◽  
Initial Energy ◽  
Blowup Time ◽  
Positive Initial Energy ◽  
Existence And Nonexistence ◽  
Nonexistence Of Solutions ◽  
Coupled Parabolic Systems

<p style='text-indent:20px;'>In this paper, we consider a class of finitely degenerate coupled parabolic systems. At high initial energy level <inline-formula><tex-math id="M1">\begin{document}$ J(u_{0})&gt;d $\end{document}</tex-math></inline-formula>, we present a new sufficient condition to describe the global existence and nonexistence of solutions for problem (1)-(4) respectively. Moreover, by applying the Levine's concavity method, we give some affirmative answers to finite time blow up of solutions at arbitrary positive initial energy <inline-formula><tex-math id="M2">\begin{document}$ J(u_{0})&gt;0 $\end{document}</tex-math></inline-formula>, including the estimate of upper bound of blowup time.</p>

scholarly journals Carleman estimates and null controllability of a class of singular parabolic equations

2018 ◽  
Vol 8 (1) ◽  
pp. 1057-1082
Author(s):  
Runmei Du ◽  
Jürgen Eichhorn ◽  
Qiang Liu ◽  
Chunpeng Wang
Keyword(s):  
Control Systems ◽  
Parabolic Equations ◽  
Null Controllability ◽  
Carleman Estimates ◽  
Energy Estimates ◽  
Semilinear Parabolic Equations ◽  
Linearized Equations ◽  
Singular Parabolic Equations ◽  
Semilinear Parabolic

Abstract In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equations.

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