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Evolution Equations & Control Theory

June 2019 , Volume 8 , Issue 2

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On a Mathematical model with non-compact boundary conditions describing bacterial population (Ⅱ)
Mohamed Boulanouar
2019, 8(2): 247-271 doi: 10.3934/eect.2019014 +[Abstract](1282) +[HTML](341) +[PDF](459.76KB)
Abstract:

This work is a natural continuation of an earlier one [1] in which a mathematical model has been studied. This model is based on maturation-velocity structured bacterial population. The bacterial mitosis is mathematically described by a non-compact boundary condition. We investigate the spectral properties of the generated semigroup and we give an explicit estimation of the bound of its infinitesimal generator.

Visualization of the convex integration solutions to the Monge-Ampère equation
Luca Codenotti and Marta Lewicka
2019, 8(2): 273-300 doi: 10.3934/eect.2019015 +[Abstract](1353) +[HTML](392) +[PDF](6468.89KB)
Abstract:

In this article, we implement the algorithm based on the convex integration result proved in [13] and obtain visualizations of the first iterations of the Nash-Kuiper scheme, approximating the anomalous solutions to the Monge-Ampère equation in two dimensions.

Generation of semigroups for the thermoelastic plate equation with free boundary conditions
Robert Denk and Yoshihiro Shibata
2019, 8(2): 301-313 doi: 10.3934/eect.2019016 +[Abstract](1332) +[HTML](362) +[PDF](417.04KB)
Abstract:

We consider the linear thermoelastic plate equations with free boundary conditions in uniform \begin{document}$ C^4 $\end{document}-domains, which includes the half-space, bounded and exterior domains. We show that the corresponding operator generates an analytic semigroup in \begin{document}$ L^p $\end{document}-spaces for all \begin{document}$ p\in(1, \infty) $\end{document} and has maximal \begin{document}$ L^q $\end{document}-\begin{document}$ L^p $\end{document}-regularity on finite time intervals. On bounded \begin{document}$ C^4 $\end{document}-domains, we obtain exponential stability.

Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data
Monika Eisenmann, Etienne Emmrich and Volker Mehrmann
2019, 8(2): 315-342 doi: 10.3934/eect.2019017 +[Abstract](1303) +[HTML](336) +[PDF](475.97KB)
Abstract:

For initial value problems associated with operator-valued Riccati differential equations posed in the space of Hilbert–Schmidt operators existence of solutions is studied. An existence result known for algebraic Riccati equations is generalized and used to obtain the existence of a solution to the approximation of the problem via a backward Euler scheme. Weak and strong convergence of the sequence of approximate solutions is established permitting a large class of right-hand sides and initial data.

Stability of the anisotropic Maxwell equations with a conductivity term
Matthias Eller
2019, 8(2): 343-357 doi: 10.3934/eect.2019018 +[Abstract](1281) +[HTML](421) +[PDF](393.31KB)
Abstract:

The dynamic Maxwell equations with a conductivity term are considered. Conditions for the exponential and strong stability of an initial-boundary value problem are given. The permeability and the permittivity are assumed to be \begin{document}$ 3\times 3 $\end{document} symmetric, positive definite tensors. A result concerning solutions of higher regularity is obtained along the way.

Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms
Vo Anh Khoa, Le Thi Phuong Ngoc and Nguyen Thanh Long
2019, 8(2): 359-395 doi: 10.3934/eect.2019019 +[Abstract](1524) +[HTML](413) +[PDF](797.91KB)
Abstract:

In this paper we consider a porous-elastic system consisting of nonlinear boundary/interior damping and nonlinear boundary/interior sources. Our interest lies in the theoretical understanding of the existence, finite time blow-up of solutions and their exponential decay using non-trivial adaptations of well-known techniques. First, we apply the conventional Faedo-Galerkin method with standard arguments of density on the regularity of initial conditions to establish two local existence theorems of weak solutions. Moreover, we detail the uniqueness result in some specific cases. In the second theme, we prove that any weak solution possessing negative initial energy has the latent blow-up in finite time. Finally, we obtain the so-called exponential decay estimates for the global solution under the construction of a suitable Lyapunov functional. In order to corroborate our theoretical decay, a numerical example is provided.

The cost of boundary controllability for a parabolic equation with inverse square potential
Patrick Martinez and Judith Vancostenoble
2019, 8(2): 397-422 doi: 10.3934/eect.2019020 +[Abstract](1235) +[HTML](383) +[PDF](538.57KB)
Abstract:

The goal of this paper is to analyze the cost of boundary null controllability for the \begin{document}$ 1-D $\end{document} linear heat equation with the so-called inverse square potential:

where \begin{document}$ \mu $\end{document} is a real parameter such that \begin{document}$ \mu \leq 1/4 $\end{document}. Since the works by Baras and Goldstein [4,5], it is known that such problems are well-posed for any \begin{document}$ \mu \leq 1/4 $\end{document} (the constant appearing in the Hardy inequality) whereas instantaneous blow-up may occur when \begin{document}$ \mu>1/4 $\end{document}. For any \begin{document}$ \mu \leq 1/4 $\end{document}, it has been proved in [52] (via Carleman estimates) that the equation can be controlled (in any time \begin{document}$ T>0 $\end{document}) by a locally distributed control. Obviously, the same result holds true when one considers the case of a boundary control acting at \begin{document}$ x = 1 $\end{document}. The goal of the present paper is to provide sharp estimates of the cost of the control in that case, analyzing its dependence with respect to the two paramaters \begin{document}$ T>0 $\end{document} and \begin{document}$ \mu \in (-\infty, 1/4] $\end{document}. Our proofs are based on the moment method and very recent results on biorthogonal sequences.

Decay rate of the Timoshenko system with one boundary damping
Denis Mercier and Virginie Régnier
2019, 8(2): 423-445 doi: 10.3934/eect.2019021 +[Abstract](1531) +[HTML](387) +[PDF](473.76KB)
Abstract:

In this paper, we study the indirect boundary stabilization of the Timoshenko system with only one dissipation law. This system, which models the dynamics of a beam, is a hyperbolic system with two wave speeds. Assuming that the wave speeds are equal, we prove exponential stability. Otherwise, we show that the decay rate is of exponential or polynomial type. Note that the results hold without the technical assumptions on the coefficients coming from the multiplier method: a sharp analysis of the behaviour of the resolvent operator along the imaginary axis is performed to avoid those artificial restrictions.

Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods
Toshiyuki Suzuki
2019, 8(2): 447-471 doi: 10.3934/eect.2019022 +[Abstract](1382) +[HTML](429) +[PDF](510.62KB)
Abstract:

We solve the scattering problems for nonlinear Schrödinger equations with an inverse-square potential by applying the energy methods. The methods are optimized to the abstract semilinear Schrödinger evolution equations with nonautonomous terms.

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