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Evolution Equations and Control Theory

June 2022 , Volume 11 , Issue 3

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Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation
Siqi Chen, Yong-Kui Chang and Yanyan Wei
2022, 11(3): 621-633 doi: 10.3934/eect.2021017 +[Abstract](1234) +[HTML](507) +[PDF](299.51KB)
Abstract:

This paper is mainly concerned with the existence of pseudo S-asymptotically Bloch type periodic solutions to damped evolution equations in Banach spaces. Some existence results for classical Cauchy conditions and nonlocal Cauchy conditions are established through properties of pseudo S-asymptotically Bloch type periodic functions and regularized families. The obtained results show that for each pseudo S-asymptotically Bloch type periodic input forcing disturbance, the output mild solutions to reference equations remain pseudo S-asymptotically Bloch type periodic.

Initial boundary value problem for a strongly damped wave equation with a general nonlinearity
Hui Yang and Yuzhu Han
2022, 11(3): 635-648 doi: 10.3934/eect.2021019 +[Abstract](1191) +[HTML](462) +[PDF](329.98KB)
Abstract:

In this paper, a strongly damped semilinear wave equation with a general nonlinearity is considered. With the help of a newly constructed auxiliary functional and the concavity argument, a general finite time blow-up criterion is established for this problem. Furthermore, the lifespan of the weak solution is estimated from both above and below. This partially extends some results obtained in recent literatures and sheds some light on the similar effect of power type nonlinearity and logarithmic nonlinearity on finite time blow-up of solutions to such problems.

Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations
Manil T. Mohan
2022, 11(3): 649-679 doi: 10.3934/eect.2021020 +[Abstract](1108) +[HTML](426) +[PDF](467.72KB)
Abstract:

The convective Brinkman-Forchheimer (CBF) equations describe the motion of incompressible viscous fluids through a rigid, homogeneous, isotropic, porous medium and is given by

In this work, we consider some distributed optimal control problems like total energy minimization, minimization of enstrophy, etc governed by the two dimensional CBF equations with the absorption exponent \begin{document}$ r = 1,2 $\end{document} and \begin{document}$ 3 $\end{document}. We show the existence of an optimal solution and the first order necessary conditions of optimality for such optimal control problems in terms of the Euler-Lagrange system. Furthermore, for the case \begin{document}$ r = 3 $\end{document}, we show the second order necessary and sufficient conditions of optimality. We also investigate an another control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is 2D CBF equations, using optimal control techniques.

Exact controllability and stabilization for a general internal wave system of Benjamin-Ono type
José R. Quintero and Alex M. Montes
2022, 11(3): 681-709 doi: 10.3934/eect.2021021 +[Abstract](1014) +[HTML](466) +[PDF](406.3KB)
Abstract:

In this work we consider the exact controllability and stabilization on a periodic domain for the generalized Benjamin-Ono type system for internal waves. The exact controllability of the linearized model is proved by using the moment method and spectral analysis. In order to get the same result for the nonlinear model, we use a fixed point argument in Sobolev spaces.

On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping
Roger P. de Moura, Ailton C. Nascimento and Gleison N. Santos
2022, 11(3): 711-727 doi: 10.3934/eect.2021022 +[Abstract](1203) +[HTML](439) +[PDF](374.29KB)
Abstract:

In this paper we prove the exponential decay of the energy for the high-order Kadomtsev-Petviashvili II equation with localized damping. To do that, we use the classical dissipation-observability method and a unique continuation principle introduced by Bourgain in [3] here extended for the high-order Kadomtsev-Petviashvili. A similar result is also obtained for the two-dimensional Zakharov-Kuznetsov (ZK)equation. The method of proof works better for the ZK equation, so we were led to make some subtle modifications on it to include KP type equations. In fact, to reach a key estimate we use an anisotropic Gagliardo-Nirenberg inequality to drop the \begin{document}$ y $\end{document}-derivative of the norm.

Exponential stability for a multi-particle system with piecewise interaction function and stochastic disturbance
Yipeng Chen, Yicheng Liu and Xiao Wang
2022, 11(3): 729-748 doi: 10.3934/eect.2021023 +[Abstract](1021) +[HTML](441) +[PDF](3068.71KB)
Abstract:

In this paper, a generalized Motsch-Tadmor model with piecewise interaction function is investigated, which can be viewed as a generalization of the model proposed in [9]. Our analysis bases on the connectedness of the underlying graph of the system. Some sufficient conditions are presented to guarantee the system to achieve flocking. Besides, we add a stochastic disturbance to the system and consider the flocking in the sense of expectation. As results, some criterions to the flocking solution with exponential convergent rate are established by the standard differential equations analysis.

Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains
André da Rocha Lopes and Juan Límaco
2022, 11(3): 749-779 doi: 10.3934/eect.2021024 +[Abstract](997) +[HTML](401) +[PDF](445.56KB)
Abstract:

In this paper, we establish a local null controllability result for a nonlinear parabolic PDE with local and nonlocal nonlinearities in a domain whose boundary moves in time by a control force with a multiplicative part acting on a prescribed subdomain. We prove that, if the initial data is sufficiently small and the linearized system at zero satisfies an appropriate condition, the equation can be driven to zero.

The lifespan of solutions for a viscoelastic wave equation with a strong damping and logarithmic nonlinearity
Menglan Liao
2022, 11(3): 781-792 doi: 10.3934/eect.2021025 +[Abstract](1077) +[HTML](406) +[PDF](326.25KB)
Abstract:

This paper deals with the following viscoelastic wave equation with a strong damping and logarithmic nonlinearity:

A finite time blow-up result is proved for high initial energy. Meanwhile, the lifespan of the weak solution is discussed. The present results in this paper complement and improve the previous work that is obtained by Ha and Park [Adv. Differ. Equ., (2020) 2020: 235].

Functional inequalities involving nonlocal operators on complete Riemannian manifolds and their applications to the fractional porous medium equation
Nikolaos Roidos and Yuanzhen Shao
2022, 11(3): 793-825 doi: 10.3934/eect.2021026 +[Abstract](1119) +[HTML](414) +[PDF](462.04KB)
Abstract:

The objective of this paper is twofold. First, we conduct a careful study of various functional inequalities involving the fractional Laplacian operators, including nonlocal Sobolev-Poincaré, Nash, Super Poincaré and logarithmic Sobolev type inequalities, on complete Riemannian manifolds satisfying some mild geometric assumptions. Second, based on the derived nonlocal functional inequalities, we analyze the asymptotic behavior of the solution to the fractional porous medium equation, \begin{document}$ \partial_t u +(-\Delta)^\sigma (|u|^{m-1}u ) = 0 $\end{document} with \begin{document}$ m>0 $\end{document} and \begin{document}$ \sigma\in (0, 1) $\end{document}. In addition, we establish the global well-posedness of the equation on an arbitrary complete Riemannian manifold.

On a class of differential quasi-variational-hemivariational inequalities in infinite-dimensional Banach spaces
Savin Treanţă
2022, 11(3): 827-836 doi: 10.3934/eect.2021027 +[Abstract](928) +[HTML](413) +[PDF](308.32KB)
Abstract:

A class of differential quasi-variational-hemivariational inequalities (DQVHI, for short) is studied in this paper. First, based on the Browder's result, KKM theorem and monotonicity arguments, we prove the superpositionally measurability, convexity and strongly-weakly upper semicontinuity for the solution set of a general quasi-variational-hemivariational inequality. Further, by using optimal control theory, measurability of set-valued mappings and the theory of semigroups, we establish that the solution set of (DQVHI) is nonempty and compact. This kind of evolutionary problems incorporates various classes of problems and models.

On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions
Phan Van Tin
2022, 11(3): 837-867 doi: 10.3934/eect.2021028 +[Abstract](999) +[HTML](405) +[PDF](409.75KB)
Abstract:

In this paper we consider the Schrödinger equation with nonlinear derivative term. Our goal is to initiate the study of this equation with non vanishing boundary conditions. We obtain the local well posedness for the Cauchy problem on Zhidkov spaces \begin{document}$ X^k( \mathbb{R}) $\end{document} and in \begin{document}$ \phi+H^k( \mathbb{R}) $\end{document}. Moreover, we prove the existence of conservation laws by using localizing functions. Finally, we give explicit formulas for stationary solutions on Zhidkov spaces.

Persistence of superoscillations under the Schrödinger equation
Elodie Pozzi and Brett D. Wick
2022, 11(3): 869-894 doi: 10.3934/eect.2021029 +[Abstract](857) +[HTML](329) +[PDF](417.26KB)
Abstract:

The goal of this paper is to provide new proofs of the persistence of superoscillations under the Schrödinger equation. Superoscillations were first put forward by Aharonov and have since received much study because of connections to physics, engineering, signal processing and information theory. An interesting mathematical question is to understand the time evolution of superoscillations under certain Schrödinger equations arising in physics. This paper provides an alternative proof of the persistence of superoscillations by some elementary convergence facts for sequence and series and some connections with certain polynomials and identities in combinatorics. The approach given opens new perspectives to establish persistence of superoscillations for the Schrödinger equation with more general potentials.

Long-time behavior of a size-structured population model with diffusion and delayed birth process
Dongxue Yan and Xianlong Fu
2022, 11(3): 895-923 doi: 10.3934/eect.2021030 +[Abstract](917) +[HTML](414) +[PDF](587.28KB)
Abstract:

This work focuses on the long time behavior for a size-dependent population system with diffusion and Riker type birth function. Some dynamical properties of the considered system is investigated by using \begin{document}$ C_0 $\end{document}-semigroup theory and spectral analysis arguments. Some sufficient conditions are obtained respectively for asymptotical stability, asynchronous exponential growth at the null equilibrium as well as Hopf bifurcation occurring at the positive steady state of the system. In the end several examples and their simulations are also provided to illustrate the achieved results.

Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion
Yousef Alnafisah and Hamdy M. Ahmed
2022, 11(3): 925-937 doi: 10.3934/eect.2021031 +[Abstract](1180) +[HTML](502) +[PDF](376.34KB)
Abstract:

In this paper, we study the existence and uniqueness of mild solutions for neutral delay Hilfer fractional integrodifferential equations with fractional Brownian motion. Sufficient conditions for controllability of neutral delay Hilfer fractional differential equations with fractional Brownian motion are established. The required results are obtained based on the fixed point theorem combined with the semigroup theory, fractional calculus and stochastic analysis. Finally, an example is given to illustrate the obtained results.

Local stabilization of viscous Burgers equation with memory
Wasim Akram and Debanjana Mitra
2022, 11(3): 939-973 doi: 10.3934/eect.2021032 +[Abstract](1051) +[HTML](353) +[PDF](533.83KB)
Abstract:

In this article, we study the local stabilization of the viscous Burgers equation with memory around the steady state zero using localized interior controls. We first consider the linearized equation around zero which corresponds to a system coupled between a parabolic equation and an ODE. We show the feedback stabilization of the system with any exponential decay \begin{document}$ -\omega $\end{document}, where \begin{document}$ \omega\in (0, \omega_0) $\end{document}, for some \begin{document}$ \omega_0>0 $\end{document}, using a finite dimensional localized interior control. The control is obtained from the solution of a suitable degenerate Riccati equation. We do an explicit analysis of the spectrum of the corresponding linearized operator. In fact, \begin{document}$ \omega_0 $\end{document} is the unique accumulation point of the spectrum of the operator. We also show that the system is not stabilizable with exponential decay \begin{document}$ -\omega $\end{document}, where \begin{document}$ \omega>\omega_0 $\end{document}, using any \begin{document}$ L^2 $\end{document}-control. Finally, we obtain the local stabilization result for the nonlinear system by means of the feedback control stabilizing the linearized system using the Banach fixed point theorem.

Solvability of doubly nonlinear parabolic equation with p-laplacian
Shun Uchida
2022, 11(3): 975-1000 doi: 10.3934/eect.2021033 +[Abstract](954) +[HTML](340) +[PDF](406.11KB)
Abstract:

In this paper, we consider a doubly nonlinear parabolic equation \begin{document}$ \partial _t \beta (u) - \nabla \cdot \alpha (x , \nabla u) \ni f $\end{document} with the homogeneous Dirichlet boundary condition in a bounded domain, where \begin{document}$ \beta : \mathbb{R} \to 2 ^{ \mathbb{R} } $\end{document} is a maximal monotone graph satisfying \begin{document}$ 0 \in \beta (0) $\end{document} and \begin{document}$ \nabla \cdot \alpha (x , \nabla u ) $\end{document} stands for a generalized \begin{document}$ p $\end{document}-Laplacian. Existence of solution to the initial boundary value problem of this equation has been studied in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on \begin{document}$ \beta $\end{document}. However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for \begin{document}$ 1 < p < 2 $\end{document}. Main purpose of this paper is to show the solvability of the initial boundary value problem for any \begin{document}$ p \in (1, \infty ) $\end{document} without any conditions for \begin{document}$ \beta $\end{document} except \begin{document}$ 0 \in \beta (0) $\end{document}. We also discuss the uniqueness of solution by using properties of entropy solution.

2021 Impact Factor: 1.169
5 Year Impact Factor: 1.294
2021 CiteScore: 2

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