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

<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>

Global and non global solutions for a class of coupled parabolic systems

In the present paper, we investigate the global well-posedness and exponential decay for some coupled non-linear heat equations. Moreover, we discuss the global and non global existence of solutions using the potential well method.

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

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.

Adaptive stabilization based on passive and swapping identifiers for a class of uncertain linearized Ginzburg–Landau equations

This paper is devoted to the stabilization for a class of uncertain linearized Ginzburg–Landau equations (GLEs). The distinguishing feature of such system is the presence of serious uncertainties which enlarge the scope of the systems whereas challenge the control problem. Therefore, certain dynamic compensation mechanisms are required to overcome the uncertainties of system. Motivated by the related literature, the original complex-valued GLEs are transformed into a class of real-valued coupled parabolic systems with serious uncertainties and distinctive characteristics. For this, two classes of identifiers respectively based on passive and swapping identifiers are first introduced to design parameter dynamic compensators. Then, by combining infinite-dimensional backstepping method with the dynamic compensators, two adaptive state-feedback controllers are constructed which guarantee all the closed-loop system states are bounded while the original system states converge to zero. A numerical example is provided to validate the effectiveness of the theoretical results.