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Evolution Equations & Control Theory

June 2021 , Volume 10 , Issue 2

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Periodic solutions and multiharmonic expansions for the Westervelt equation
Barbara Kaltenbacher
2021, 10(2): 229-247 doi: 10.3934/eect.2020063 +[Abstract](527) +[HTML](291) +[PDF](436.75KB)

In this paper we consider nonlinear time periodic sound propagation according to the Westervelt equation, which is a classical model of nonlinear acoustics and a second order quasilinear strongly damped wave equation exhibiting potential degeneracy. We prove existence, uniqueness and regularity of solutions with time periodic forcing and time periodic initial-end conditions, on a bounded domain with absorbing boundary conditions. In order to mathematically recover the physical phenomenon of higher harmonics, we expand the solution as a superposition of contributions at frequencies that are multiples of a fundamental excitation frequency. This multiharmonic expansion is proven to converge, in appropriate function spaces, to the periodic solution in time domain.

Decay rate of global solutions to three dimensional generalized MHD system
Yanxia Niu, Yinxia Wang and Qingnian Zhang
2021, 10(2): 249-258 doi: 10.3934/eect.2020064 +[Abstract](508) +[HTML](272) +[PDF](387.27KB)

We investigate the initial value problem for the three dimensional generalized incompressible MHD system. Analyticity of global solutions was proved by energy method in the Fourier space and continuous argument. Then decay rate of global small solutions in the function space \begin{document}$ \mathcal {X}^{1-2\alpha}\bigcap \mathcal {X}^{1-2\beta} $\end{document} follows by constructing time weighted energy inequality.

Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy
Ge Zu and Bin Guo
2021, 10(2): 259-270 doi: 10.3934/eect.2020065 +[Abstract](596) +[HTML](284) +[PDF](375.86KB)

The main aim of this paper is to deal with the upper and lower bounds for blow-up time of solutions to the following equation:

which has been studied in [5]. For high initial energy, it is well known that the classical potential well method is not effective. In order to overcome this difficulty, the authors apply the new energy estimate method to establish the lower bound of the \begin{document}$ L^{2}(\Omega) $\end{document} norm of the solution. Furthermore, the authors construct a new control functional and combine energy inequalities with the concavity argument to prove that the solution blows up in finite time for high initial energy. Meanwhile, an estimate of the upper bound of blow-up time is also obtained. Finally, a lower bound for blow-up time is obtained by introducing a new control functional. These results fill the gap of [5].

On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay
V. Vijayakumar, R. Udhayakumar and K. Kavitha
2021, 10(2): 271-396 doi: 10.3934/eect.2020066 +[Abstract](893) +[HTML](396) +[PDF](461.9KB)

In our manuscript, we organize a group of sufficient conditions of neutral integro-differential inclusions of Sobolev-type with infinite delay via resolvent operators. By applying Bohnenblust-Karlin's fixed point theorem for multivalued maps, we proved our results. Lastly, we present an application to support the validity of the study.

Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting
Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang and Elvira Zappale
2021, 10(2): 297-320 doi: 10.3934/eect.2020067 +[Abstract](624) +[HTML](294) +[PDF](503.42KB)

The \begin{document}$ \Gamma $\end{document}-limit of a family of functionals \begin{document}$ u\mapsto \int_{\Omega }f\left( \frac{x}{\varepsilon },\frac{x}{\varepsilon ^{2}},D^{s}u\right) dx $\end{document} is obtained for \begin{document}$ s = 1,2 $\end{document} and when the integrand \begin{document}$ f = f\left( y,z,v\right) $\end{document} is a continous function, periodic in \begin{document}$ y $\end{document} and \begin{document}$ z $\end{document} and convex with respect to \begin{document}$ v $\end{document} with nonstandard growth. The reiterated two-scale limits of second order derivatives are characterized in this setting.

Stabilization of the transmission wave/plate equation with variable coefficients on $ {\mathbb{R}}^n $
Bei Gong, Zhen-Hu Ning and Fengyan Yang
2021, 10(2): 321-331 doi: 10.3934/eect.2020068 +[Abstract](651) +[HTML](309) +[PDF](384.86KB)

In this article, we consider the transmission wave/plate equation with variable coefficients on \begin{document}$ {\mathbb{R}}^n(n\ge 3) $\end{document}. By virtue of the Morawetz multipliers in non Euclidean geometries and compactness-uniqueness arguments, we obtain some stability result of the transmission wave/plate system under suitable geometric conditions.

Internal feedback stabilization for parabolic systems coupled in zero or first order terms
Elena-Alexandra Melnig
2021, 10(2): 333-351 doi: 10.3934/eect.2020069 +[Abstract](520) +[HTML](264) +[PDF](420.09KB)

We consider systems of \begin{document}$ n $\end{document} parabolic equations coupled in zero or first order terms with \begin{document}$ m $\end{document} scalar controls acting through a control matrix \begin{document}$ B $\end{document}. We are interested in stabilization with a control in feedback form. Our approach relies on the approximate controllability of the linearized system, which in turn is related to unique continuation property for the adjoint system. For the unique continuation we establish algebraic Kalman type conditions.

Homogenization of a stochastic viscous transport equation
Ioana Ciotir, Nicolas Forcadel and Wilfredo Salazar
2021, 10(2): 353-364 doi: 10.3934/eect.2020070 +[Abstract](499) +[HTML](265) +[PDF](481.9KB)

In the present paper we prove an homogenisation result for a locally perturbed transport stochastic equation. The model is similar to the stochastic Burgers' equation and it is inspired by the LWR model. Therefore, the interest in studying this equation comes from it's application for traffic flow modelling. In the first part of paper we study the inhomogeneous equation. More precisely we give an existence and uniqueness result for the solution. The technical difficulties of this part come from the presence of the function \begin{document}$ \varphi $\end{document} under assumptions coherent for the model, which is giving the inhomogeneity with respect to the space variable, not present in the classical results. The second part of the paper is the homogenisation result in space.

Stability and dynamics for a nonlinear one-dimensional full compressible non-Newtonian fluids
Xin Liu, Yongjin Lu and Xin-Guang Yang
2021, 10(2): 365-384 doi: 10.3934/eect.2020071 +[Abstract](562) +[HTML](268) +[PDF](516.67KB)

In this paper, we concern with the existence of global attractors for a one-dimensional full compressible non-Newtonian fluid model defined on bounded domain. Using some delicate regular estimates and energy functional to obtain the continuity of semigroup and dissipation respectively, the long time behavior of global solution has been investigated, which is a further of [31].

Well-posedness of infinite-dimensional non-autonomous passive boundary control systems
Birgit Jacob and Hafida Laasri
2021, 10(2): 385-409 doi: 10.3934/eect.2020072 +[Abstract](604) +[HTML](278) +[PDF](606.68KB)

We study a class of non-autonomous linear boundary control and observation systems that are governed by non-autonomous multiplicative perturbations. This class is motivated by fundamental partial differential equations, such as controlled wave equations and Timoshenko beams. Our main results give sufficient condition for well-posedness, existence and uniqueness of classical and mild solutions.

Approximation theorems for controllability problem governed by fractional differential equation
Rajesh Dhayal, Muslim Malik, Syed Abbas, Anil Kumar and Rathinasamy Sakthivel
2021, 10(2): 411-429 doi: 10.3934/eect.2020073 +[Abstract](773) +[HTML](355) +[PDF](511.51KB)

In this manuscript, we discuss the optimal control problem for a nonlinear system governed by the fractional differential equation in a separable Hilbert space \begin{document}$ X $\end{document}. We utilize the fixed point technique and \begin{document}$ \eta $\end{document}-resolvent family to present the existence of control for the fractional system. The optimal pair is obtained as the limit of the optimal pair sequence of the unconstrained problem. Further, we derive some approximation results, which guarantee the convergence of the numerical method to optimal pair sequence. Finally, the main results are validated with the aid of an example.

2019  Impact Factor: 0.953



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