# American Institute of Mathematical Sciences

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2163-2480

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## Evolution Equations & Control Theory

December 2015 , Volume 4 , Issue 4

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2015, 4(4): 373-389 doi: 10.3934/eect.2015.4.373 +[Abstract](753) +[PDF](481.8KB)
Abstract:
In this paper, an abstract nonsimple thermoelastic problem involving higher order gradients of displacement is considered with Dirichlet boundary conditions. We prove that the linear operator of the proposed system generates a strongly continuous semigroup which decays exponentially to zero. The optimal decay rate is determined explicitly by the physical parameters of the problem. Then we show the approximate controllability of the considered problem.
2015, 4(4): 391-429 doi: 10.3934/eect.2015.4.391 +[Abstract](1033) +[PDF](698.8KB)
Abstract:
We give relatively simple sufficient conditions on a Fourier multiplier so that it maps functions with the Hölder property with respect to a part of the variables to functions with the Hölder property with respect to all variables. By using these these sufficient conditions we prove solvability in Hölder classes of the initial-boundary value problems for the linearized Cahn-Hilliard equation with dynamic boundary conditions of two types. In addition, Schauders estimates are derived for the solutions corresponding to the problem under study.
2015, 4(4): 431-445 doi: 10.3934/eect.2015.4.431 +[Abstract](1164) +[PDF](441.7KB)
Abstract:
In this paper, we undertake a comprehensive study for the Schrödinger-Hartree equation \begin{equation*} iu_t +\Delta u+ \lambda (I_\alpha \ast |u|^{p})|u|^{p-2}u=0, \end{equation*} where $I_\alpha$ is the Riesz potential. Firstly, we address questions related to local and global well-posedness, finite time blow-up. Secondly, we derive the best constant of a Gagliardo-Nirenberg type inequality. Thirdly, the mass concentration is established for all the blow-up solutions in the $L^2$-critical case. Finally, the dynamics of the blow-up solutions with critical mass is in detail investigated in terms of the ground state.
2015, 4(4): 447-491 doi: 10.3934/eect.2015.4.447 +[Abstract](1396) +[PDF](724.1KB)
Abstract:
The aim of this paper is to highlight some recent developments and outcomes in the mathematical analysis of partial differential equations describing nonlinear sound propagation. Here the emphasis lies on well-posedness and decay results, first of all for the classical models of nonlinear acoustics, later on also for some higher order models. Besides quoting results, we also try to give an idea on their derivation by showning some of the crucial energy estimates. A section is devoted to optimization problems arising in the practical use of high intensity ultrasound.
While this review puts a certain focus on results obtained in the context of the mentioned FWF project, we also provide some important additional references (although certainly not all of them) for interesting further reading.
2015, 4(4): 493-505 doi: 10.3934/eect.2015.4.493 +[Abstract](853) +[PDF](429.2KB)
Abstract:
General nonlinear time-varying differential systems with delay are considered. Several new explicit criteria for exponential stability are given. A discussion of the obtained results and two illustrative examples are presented.
2015, 4(4): 507-524 doi: 10.3934/eect.2015.4.507 +[Abstract](1655) +[PDF](424.8KB)
Abstract:
In this paper, we study the existence and controllability for fractional evolution inclusions in Banach spaces. We use a new approach to obtain the existence of mild solutions and controllability results, avoiding hypotheses of compactness on the semigroup generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. Finally, two examples are given to illustrate our theoretical results.

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