On an inverse problem for fractional evolution equation
Nguyen Huy Tuan Mokhtar Kirane Long Dinh Le Van Thinh Nguyen
Evolution Equations & Control Theory 2017, 6(1): 111-134 doi: 10.3934/eect.2017007

In this paper, we investigate a backward problem for a fractional abstract evolution equation for which we wants to extract the initial distribution from the observation data provided along the final time $t = T.$ This problem is well-known to be ill-posed due to the rapid decay of the forward process. We consider a final value problem for fractional evolution process with respect to time. For this ill-posed problem, we construct two regularized solutions using quasi-reversibility method and quasi-boundary value method. The well-posedness of the regularized solutions as well as the convergence property is analyzed. The advantage of the proposed methods is that the regularized solution is given analytically and therefore is easy to be implemented. A numerical example is presented to show the validity of the proposed methods.

keywords: Fractional evolution equation backward problem regularization
Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks
Mokhtar Kirane Belkacem Said-Houari Mohamed Naim Anwar
Communications on Pure & Applied Analysis 2011, 10(2): 667-686 doi: 10.3934/cpaa.2011.10.667
We study the exponential stability of the Timoshenko beam system by interior time-dependent delay term feedbacks. The beam is clamped at the two hand points subject to two internal feedbacks: one with a time-varying delay and the other without delay. Using the variable norm technique of Kato, it is proved that the system is well-posed whenever an hypothesis between the weight of the delay term in the feedback, the weight of the term without delay and the wave speeds. By introducing an appropriate Lyapunov functional the exponential stability of the system is proved. Under the imposed constrain on the weights of the feedbacks and the wave speeds, the exponential decay of the energy is established via a suitable Lyapunov functional.
keywords: global solutions damping delay exponential decay. stability Timoshenko
Determination of initial data for a reaction-diffusion system with variable coefficients
Vo Van Au Mokhtar Kirane Nguyen Huy Tuan
Discrete & Continuous Dynamical Systems - A 2019, 39(2): 771-801 doi: 10.3934/dcds.2019032

In this paper, we study a final value problem for a reaction-diffusion system with time and space dependent diffusion coefficients. In general, the inverse problem of identifying the initial data is not well-posed, and herein the Hadamard-instability occurs. Applying a new version of a modified quasi-reversibility method, we propose a stable approximate (regularized) problem. The existence, uniqueness and stability of the corresponding regularized problem are obtained. Furthermore, we also investigate the error estimate and show that the approximate solution converges to the exact solution in $L_2$ and $\stackrel{0}{H_1}$ norms. Our method can be applied to some concrete models that arise in biology, chemical engineering, etc.

keywords: Backward problem reaction-diffusion system ill-posed problem quasi-reversibility method regularization

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