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Evolution Equations & Control Theory

2013 , Volume 2 , Issue 2

Special issue on Nonlinear

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Preface: Introduction to the Special Volume on Nonlinear PDEs and Control Theory with Applications
Lorena Bociu, Barbara Kaltenbacher and Petronela Radu
2013, 2(2): i-ii doi: 10.3934/eect.2013.2.2i +[Abstract](72) +[PDF](93.4KB)
This volume collects a number of contributions in the fields of partial differential equations and control theory, following the Special Session Nonlinear PDEs and Control Theory with Applications held at the 9th AIMS conference on Dynamical Systems, Differential Equations and Applications in Orlando, July 1--5, 2012.

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Stability analysis of non-linear plates coupled with Darcy flows
Eugenio Aulisa, Akif Ibragimov and Emine Yasemen Kaya-Cekin
2013, 2(2): 193-232 doi: 10.3934/eect.2013.2.193 +[Abstract](94) +[PDF](606.1KB)
In this paper we study the dynamical response of a non-linear plate with viscous damping perturbed in both vertical and axial directions and interacting with Darcy flow. We first consider the problem for non-linear elastic body with damping coefficient; existence and uniqueness of the solution for the steady state problem is proven. The stability of the dynamical non-linear plate problem under certain conditions on the applied loads is investigated. Second, we explore the fluid structure interaction problem with Darcy flow in porous media. Energy functional for the displacement field of the plate and the gradient pressure of the fluid flow is built in an appropriate Sobolev type norm. We show that for a class of boundary conditions the energy functional is limited by the flux of mass through the inlet boundary.
Rational decay rates for a PDE heat--structure interaction: A frequency domain approach
George Avalos and Roberto Triggiani
2013, 2(2): 233-253 doi: 10.3934/eect.2013.2.233 +[Abstract](138) +[PDF](610.3KB)
In this paper, we consider a simplified version of a fluid--structure PDE model ---in fact, a heat--structure interaction PDE-model. It is intended to be a first step toward a more realistic fluid--structure PDE model which has been of longstanding interest within the mathematical and biological sciences [33, p. 121], [17], [19]. This physically more sound and mathematically more challenging model will be treated in [13]. The simplified model replaces the linear dynamic Stokes equation with a linear $n$-dimensional heat equation (heat--structure interaction). The entire dynamics manifests both hyperbolic and parabolic features. Our main result is as follows: Given smooth initial data---i.e., data in the domain of the associated semigroup generator---the corresponding solutions decay at the rate $o( t^{-\frac{1}{2}}) $ (see Theorem 1.3 below). The basis of our proof is the recently derived resolvent criterion in [15]. In order to apply it, however, suitable PDE-estimates need to be established for each component by also making critical use of the interface conditions. A companion paper [6] will sharpen Lemma 5.8 of the present work by use of a lengthy and technical microlocal argument as in [26,29,30,31], to obtain the optimal value $\alpha =1$; hence, the optimal decay rate $o(t^{-1})$. See Remarks 1.2,1.3.
Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping
Lorena Bociu, Petronela Radu and Daniel Toundykov
2013, 2(2): 255-279 doi: 10.3934/eect.2013.2.255 +[Abstract](75) +[PDF](560.5KB)
We study regular solutions to wave equations with super-critical source terms, e.g., of exponent $p>5$ in 3D. Such sources have been a major challenge in the investigation of finite-energy ($H^1 \times L^2$) solutions to wave PDEs for many years. The wellposedness has been settled in part, but even the local existence, for instance, in 3 dimensions requires the relation $p\leq 6m/(m+1)$ between the exponents $p$ of the source and $m$ of the viscous damping.
    We prove that smooth initial data ($H^2 \times H^1$) yields regular solutions that do not depend on the above correlation. Local existence is demonstrated for any source exponent $p\geq 1$ and any monotone damping including feedbacks growing exponentially or logarithmically at infinity, or with no damping at all. The result holds in dimensions 3 and 4, and with some restrictions on $p$ in dimensions $n\geq 5$. Furthermore, if we assert the classical condition that the damping grows as fast as the source, then these regular solutions are global.
Avoiding degeneracy in the Westervelt equation by state constrained optimal control
Christian Clason and Barbara Kaltenbacher
2013, 2(2): 281-300 doi: 10.3934/eect.2013.2.281 +[Abstract](65) +[PDF](560.5KB)
The Westervelt equation, which describes nonlinear acoustic wave propagation in high intensity ultrasound applications, exhibits potential degeneracy for large acoustic pressure values. While well-posedness results on this PDE have so far been based on smallness of the solution in a higher order spatial norm, non-degeneracy can be enforced explicitly by a pointwise state constraint in a minimization problem, thus allowing for pressures with large gradients and higher-order derivatives, as is required in the mentioned applications. Using regularity results on the linearized state equation, well-posedness and necessary optimality conditions for the PDE constrained optimization problem can be shown via a relaxation approach by Alibert and Raymond [2].
Higher differentiability in the context of Besov spaces for a class of nonlocal functionals
Mikil Foss and Joe Geisbauer
2013, 2(2): 301-318 doi: 10.3934/eect.2013.2.301 +[Abstract](84) +[PDF](387.0KB)
The aim of this paper is to contribute to the nonlocal theory within the calculus of variations by studying two classes of nonlocal functionals. Since the nonlocal theory is not quite as developed as the local theory, a proof for the existence and uniqueness of minimizers is provided. However, the main result within the paper establishes the higher differentiability, in the context of Besov spaces, for minimizers of nonlocal functionals. This result is obtained under quadratic growth assumptions via the difference quotient method.
Global existence for exterior problems of semilinear wave equations with the null condition in $2$D
Hideo Kubo
2013, 2(2): 319-335 doi: 10.3934/eect.2013.2.319 +[Abstract](74) +[PDF](431.6KB)
In this paper we deal with the exterior problem for a system of nonlinear wave equations in two space dimensions under some geometric restriction on the obstacle. We prove a global existence result for the problem with small and smooth initial data, provided that the nonlinearity is taken to be cubic and satisfies the null condition.
A local existence result for a system of viscoelasticity with physical viscosity
Marta Lewicka and Piotr B. Mucha
2013, 2(2): 337-353 doi: 10.3934/eect.2013.2.337 +[Abstract](110) +[PDF](455.9KB)
We prove the local in time existence of regular solutions to the system of equations of isothermal viscoelasticity with clamped boundary conditions. We deal with a general form of viscous stress tensor $\mathcal{Z}(F,\dot F)$, assuming a Korn-type condition on its derivative $D_{\dot F}\mathcal{Z}(F, \dot F)$. This condition is compatible with the balance of angular momentum, frame invariance and the Claussius-Duhem inequality. We give examples of linear and nonlinear (in $\dot F$) tensors $\mathcal{Z}$ satisfying these required conditions.
Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement
Shitao Liu
2013, 2(2): 355-364 doi: 10.3934/eect.2013.2.355 +[Abstract](56) +[PDF](312.3KB)
We consider an inverse problem of recovering simultaneously the sound speed and an initial condition for the wave equation from a single Dirichlet data measured on the boundary of the support of the initial condition. The problem is motived from the recently developed hybrid imaging models as well as from classical inverse hyperbolic problems with a single boundary measurement formulation. We establish uniqueness of the recovery and the proof is based on the Carleman estimate and continuous observability inequality for general Riemannian wave equations.
Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces
Stefan Meyer and Mathias Wilke
2013, 2(2): 365-378 doi: 10.3934/eect.2013.2.365 +[Abstract](59) +[PDF](415.0KB)
We investigate a quasilinear initial-boundary value problem for Kuznetsov's equation with non-homogeneous Dirichlet boundary conditions. This is a model in nonlinear acoustics which describes the propagation of sound in fluidic media with applications in medical ultrasound. We prove that there exists a unique global solution which depends continuously on the sufficiently small data and that the solution and its temporal derivatives converge at an exponential rate as time tends to infinity. Compared to the analysis of Kaltenbacher & Lasiecka, we require optimal regularity conditions on the data and give simplified proofs which are based on maximal $L_p$-regularity for parabolic equations and the implicit function theorem.
Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system
Dugan Nina, Ademir Fernando Pazoto and Lionel Rosier
2013, 2(2): 379-402 doi: 10.3934/eect.2013.2.379 +[Abstract](76) +[PDF](455.5KB)
This paper is concerned with the exact controllability problem for a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to one-dimensional horizontal motion. We take as fluid model a Boussinesq system of KdV-KdV type, and as control the acceleration of the tank. We derive for the linearized system an exact controllability result in small time in an appropriate space.
Nonlinear instability of solutions in parabolic and hyperbolic diffusion
Stephen Pankavich and Petronela Radu
2013, 2(2): 403-422 doi: 10.3934/eect.2013.2.403 +[Abstract](70) +[PDF](436.9KB)
We consider semilinear evolution equations of the form $a(t)\partial_{tt}u + b(t) \partial_t u + Lu = f(x,u)$ and $b(t) \partial_t u + Lu = f(x,u),$ with possibly unbounded $a(t)$ and possibly sign-changing damping coefficient $b(t)$, and determine precise conditions for which linear instability of the steady state solutions implies nonlinear instability. More specifically, we prove that linear instability with an eigenfunction of fixed sign gives rise to nonlinear instability by either exponential growth or finite-time blow-up. We then discuss a few examples to which our main theorem is immediately applicable, including evolution equations with supercritical and exponential nonlinearities.
Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III
Belkacem Said-Houari and Radouane Rahali
2013, 2(2): 423-440 doi: 10.3934/eect.2013.2.423 +[Abstract](99) +[PDF](439.4KB)
In this paper, we investigate the decay property of a Timoshenko system in thermoelasticity of type III in the whole space where the heat conduction is given by the Green and Naghdi theory. Surprisingly, we show that the coupling of the Timoshenko system with the heat conduction of Green and Naghdi's theory slows down the decay of the solution. In fact we show that the $L^2$-norm of the solution decays like $(1+t)^{-1/8}$, while in the case of the coupling of the Timoshenko system with the Fourier or Cattaneo heat conduction, the decay rate is of the form $(1+t)^{-1/4}$ [25]. We point out that the decay rate of $(1+t)^{-1/8}$ has been obtained provided that the initial data are in $L^1( \mathbb{R})\cap H^s(\mathbb{R}), (s\geq 2)$. If the wave speeds of the first two equations are different, then the decay rate of the solution is of regularity-loss type, that is in this case the previous decay rate can be obtained only under an additional regularity assumption on the initial data. In addition, by restricting the initial data to be in $H^{s}\left( \mathbb{R}\right)\cap L^{1,\gamma }\left( \mathbb{R}\right) $ with $ \gamma \in \left[ 0,1\right] $, we can derive faster decay estimates with the decay rate improvement by a factor of $t^{-\gamma/4}$.

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