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Evolution Equations & Control Theory

September 2021 , Volume 10 , Issue 3

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Decay rates for the Moore-Gibson-Thompson equation with memory
Hizia Bounadja and Belkacem Said Houari
2021, 10(3): 431-460 doi: 10.3934/eect.2020074 +[Abstract](1022) +[HTML](408) +[PDF](602.12KB)
Abstract:

The main goal of this paper is to investigate the existence and stability of the solutions for the Moore–Gibson–Thompson equation (MGT) with a memory term in the whole spaces \begin{document}$ \mathbb{R}^{N} $\end{document}. The MGT equation arises from modeling high-frequency ultrasound waves as an alternative model to the well-known Kuznetsov's equation. First, following [8] and [26], we show that the problem is well-posed under an appropriate assumption on the coefficients of the system. Then, we built some Lyapunov functionals by using the energy method in Fourier space. These functionals allows us to get control estimates on the Fourier image of the solution. These estimates of the Fourier image together with some integral inequalities lead to the decay rate of the \begin{document}$ L^{2} $\end{document}-norm of the solution. We use two types of memory term here: type Ⅰ memory term and type Ⅲ memory term. Decay rates are obtained in both types. More precisely, decay rates of the solution are obtained depending on the exponential or polynomial decay of the memory kernel. More importantly, we show stability of the solution in both cases: a subcritical range of the parameters and a critical range. However for the type Ⅰ memory we show in the critical case that the solution has the regularity-loss property.

A remark on the attainable set of the Schrödinger equation
Jonas Lampart
2021, 10(3): 461-469 doi: 10.3934/eect.2020075 +[Abstract](768) +[HTML](383) +[PDF](366.34KB)
Abstract:

We discuss the set of wavefunctions \begin{document}$ \psi_V(t) $\end{document} that can be obtained from a given initial condition \begin{document}$ \psi_0 $\end{document} by applying the flow of the Schrödinger operator \begin{document}$ -\Delta + V(t,x) $\end{document} and varying the potential \begin{document}$ V(t,x) $\end{document}. We show that this set has empty interior, both as a subset of the sphere in \begin{document}$ L^2( \mathbb{R}^d) $\end{document} and as a set of trajectories.

Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations
Pengyu Chen and Xuping Zhang
2021, 10(3): 471-489 doi: 10.3934/eect.2020076 +[Abstract](996) +[HTML](474) +[PDF](525.74KB)
Abstract:

In this paper, we are considered with approximate controllability for a class of non-autonomous stochastic evolution equations of parabolic type with discrete nonlocal initial conditions. Some new results about existence of mild solutions as well as approximate controllability are established under more natural conditions on nonlinear functions and control operator by introducing a new Green function and using the theory of evolution family, Schauder fixed point theorem and the resolvent operator condition. At last, as a sample of application, these results are applied to a class of non-autonomous stochastic partial differential equation of parabolic type with discrete nonlocal initial conditions. The results obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

New results on controllability of fractional evolution systems with order $ \alpha\in (1,2) $
Yong Zhou and Jia Wei He
2021, 10(3): 491-509 doi: 10.3934/eect.2020077 +[Abstract](1123) +[HTML](411) +[PDF](441.62KB)
Abstract:

This paper addresses some interesting results of mild solutions to fractional evolution systems with order \begin{document}$ \alpha\in (1,2) $\end{document} in Banach spaces as well as the controllability problem. Firstly, we deduce a new representation of solution operators and give a new concept of mild solutions for the objective equations by the Laplace transform and Mainardi's Wright-type function, and then we proceed to establish a new compact result of the solution operators when the sine family is compact. Secondly, the controllability results of mild solutions are obtained. Finally, an example is presented to illustrate the main results.

Conditional regularity for the 3D Navier-Stokes equations in terms of the middle eigenvalue of the strain tensor
Fan Wu
2021, 10(3): 511-518 doi: 10.3934/eect.2020078 +[Abstract](846) +[HTML](377) +[PDF](362.69KB)
Abstract:

In this paper, we consider the regularity criteria for the 3D incompressible Navier-Stokes equations involving the middle eigenvalue (\begin{document}$ \lambda_2 $\end{document}) of the strain tensor. It is proved that if \begin{document}$ \lambda^+_2 $\end{document} belongs to Multiplier space or Besov space, then the weak solution remains smooth on \begin{document}$ [0, T] $\end{document}, where \begin{document}$ \lambda^{+}_2 = \max\{\lambda_2, 0\} $\end{document}. These regularity conditions allows us to improve the result obtained by Miller [7].

Funnel control for boundary control systems
Marc Puche, Timo Reis and Felix L. Schwenninger
2021, 10(3): 519-544 doi: 10.3934/eect.2020079 +[Abstract](785) +[HTML](373) +[PDF](1442.05KB)
Abstract:

We study a nonlinear, non-autonomous feedback controller applied to boundary control systems. Our aim is to track a given reference signal with prescribed performance. Existence and uniqueness of solutions to the resulting closed-loop system is proved by using nonlinear operator theory. We apply our results to both hyperbolic and parabolic equations.

Null controllability for singular cascade systems of $ n $-coupled degenerate parabolic equations by one control force
Brahim Allal, Abdelkarim Hajjaj, Lahcen Maniar and Jawad Salhi
2021, 10(3): 545-573 doi: 10.3934/eect.2020080 +[Abstract](782) +[HTML](368) +[PDF](510.83KB)
Abstract:

In this paper, we consider a class of cascade systems of \begin{document}$ n $\end{document}-coupled degenerate parabolic equations with singular lower order terms. We assume that both degeneracy and singularity occur in the interior of the space domain and we focus on null controllability problem. To this aim, we prove first Carleman estimates for the associated adjoint problem, then, we infer from it an indirect observability inequality. As a consequence, we deduce null controllability result when a unique distributed control is exerted on the system.

On finite Morse index solutions of higher order fractional elliptic equations
Belgacem Rahal and Cherif Zaidi
2021, 10(3): 575-597 doi: 10.3934/eect.2020081 +[Abstract](747) +[HTML](388) +[PDF](547.04KB)
Abstract:

We consider sign-changing solutions of the equation

where \begin{document}$ n\geq 1 $\end{document}, \begin{document}$ \lambda>0 $\end{document}, \begin{document}$ p>1 $\end{document} and \begin{document}$ 1<s\leq2 $\end{document}. The main goal of this work is to analyze the influence of the linear term \begin{document}$ \lambda u $\end{document}, in order to classify stable solutions possibly unbounded and sign-changing. We prove Liouville type theorems for stable solutions or solutions which are stable outside a compact set of \begin{document}$ \mathbb R^n $\end{document}. We first derive a monotonicity formula for our equation. After that, we provide integral estimate from stability which combined with Pohozaev-type identity to obtain nonexistence results in the subcritical case with the restrictive condition \begin{document}$ |u|_{L^{\infty}( \mathbb R^n)}^{p-1}< \frac{\lambda (p+1) }{2} $\end{document}. The supercritical case needs more involved analysis, motivated by the monotonicity formula, we then reduce the nonexistence of nontrivial entire solutions which are stable outside a compact set of \begin{document}$ \mathbb R^n $\end{document}. Through this approach we give a complete classification of stable solutions for all \begin{document}$ p>1 $\end{document}. Moreover, for the case \begin{document}$ 0<s\leq1 $\end{document}, finite Morse index solutions are classified in [19,25].

Blow-up criteria for linearly damped nonlinear Schrödinger equations
Van Duong Dinh
2021, 10(3): 599-617 doi: 10.3934/eect.2020082 +[Abstract](926) +[HTML](449) +[PDF](459.4KB)
Abstract:

We consider the Cauchy problem for linearly damped nonlinear Schrödinger equations

where \begin{document}$ a>0 $\end{document} and \begin{document}$ \alpha>0 $\end{document}. We prove the global existence and scattering for a sufficiently large damping parameter in the energy-critical case. We also prove the existence of finite time blow-up \begin{document}$ H^1 $\end{document} solutions to the focusing problem in the mass-critical and mass-supercritical cases.

Results on controllability of non-densely characterized neutral fractional delay differential system
Kasthurisamy Jothimani, Kalimuthu Kaliraj, Sumati Kumari Panda, Kotakkaran Sooppy Nisar and Chokkalingam Ravichandran
2021, 10(3): 619-631 doi: 10.3934/eect.2020083 +[Abstract](899) +[HTML](431) +[PDF](396.43KB)
Abstract:

This work establishes the controllability of nondense fractional neutral delay differential equation under Hille-Yosida condition in Banach space. The outcomes are derived with the aid of fractional calculus theory, semigroup operator theory and Schauder fixed point theorem. Theoretical results are verified through illustration.

A nonlocal Weickert type PDE applied to multi-frame super-resolution
Fatimzehrae Ait Bella, Aissam Hadri, Abdelilah Hakim and Amine Laghrib
2021, 10(3): 633-655 doi: 10.3934/eect.2020084 +[Abstract](743) +[HTML](446) +[PDF](5141.43KB)
Abstract:

In this paper, we propose a nonlocal Weickert type PDE for the multiframe super-resolution task. The proposed PDE can not only preserve singularities and edges while smoothing, but also can keep safe the texture much better. This PDE is based on the nonlocal setting of the anisotropic diffusion behavior by constructing a nonlocal term of Weickert type, which is known by its coherence enhancing diffusion tensor properties. A mathematical study concerning the well-posedness of the nonlocal PDE is also investigated with an appropriate choice of the functional space. This PDE has demonstrated its efficiency by combining the diffusion process of Perona-Malik in the flat regions and the anisotropic diffusion of the Weickert model near strong edges, as well as the ability of the non-local term to preserve the texture. The elaborated experimental results give a great insight into the effectiveness of the proposed nonlocal PDE compared to some PDEs, visually and quantitatively.

On some damped 2 body problems
Alain Haraux
2021, 10(3): 657-671 doi: 10.3934/eect.2021009 +[Abstract](314) +[HTML](176) +[PDF](346.45KB)
Abstract:

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear which were investigated thoroughly by R. Ortega and his co-authors between 2014 and 2017, in particular all solutions are bounded and tend to \begin{document}$ 0 $\end{document} for \begin{document}$ t $\end{document} large, some of them with asymptotically spiraling exponentially fast convergence to the center. We provide explicit estimates for the bounds in the general case that we refine under specific restrictions on the initial state, and we give a formal calculation which could be used to determine practically some special asymptotically spiraling orbits. Besides, a related model with exponentially damped central charge or mass gives some explicit exponentially decaying solutions which might help future investigations. An atomic contraction hypothesis related to the asymptotic dying off of solutions proven for the dissipative model might give a solution to some intriguing phenomena observed in paleontology, familiar electrical devices and high scale cosmology.

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