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Discrete & Continuous Dynamical Systems - A

2008 , Volume 22 , Issue 4

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Preface
Gisèle Ruiz Goldstein, Jerome A. Goldstein and Alain Miranville
2008, 22(4): i-ii doi: 10.3934/dcds.2008.22.4i +[Abstract](212) +[PDF](44.3KB)
Abstract:
This special issue consists of invited and carefully refereed papers on specific topics related to evolution equations, semigroup theory and related problems. Indeed, we thought that it would be very valuable to produce such a volume on important and active areas of research.
Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface
George Avalos and Roberto Triggiani
2008, 22(4): 817-833 doi: 10.3934/dcds.2008.22.817 +[Abstract](318) +[PDF](293.2KB)
Abstract:
We consider a coupled parabolic-hyperbolic PDE system arising in fluid-structure interaction, with boundary dissipation at the interface, $d = 2,3$. Such a system is semigroup well-posed in the natural energy space [4]. We then establish that it is also uniformly (exponentially) stable, thus complementing the strong stability results of the undamped case [2], [4].
Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping
Lorena Bociu and Irena Lasiecka
2008, 22(4): 835-860 doi: 10.3934/dcds.2008.22.835 +[Abstract](278) +[PDF](454.4KB)
Abstract:
We consider finite energy solutions of a wave equation with supercritical nonlinear sources and nonlinear damping. A distinct feature of the model under consideration is the presence of nonlinear sources on the boundary driven by Neumann boundary conditions. Since Lopatinski condition fails to hold (unless the $\text{dim} (\Omega) = 1$), the analysis of the nonlinearities supported on the boundary, within the framework of weak solutions, is a rather subtle issue and involves the strong interaction between the source and the damping. Thus, it is not surprising that existence theory for this class of problems has been established only recently. However, the uniqueness of weak solutions was declared an open problem. The main result in this work is uniqueness of weak solutions. This result is proved for the same (even larger) class of data for which existence theory holds. In addition, we prove that weak solutions are continuously depending on initial data and that the flow corresponding to weak and global solutions is a dynamical system on the finite energy space.
Asymptotic behavior of population dynamics models with nonlocal distributed delays
Cecilia Cavaterra and Maurizio Grasselli
2008, 22(4): 861-883 doi: 10.3934/dcds.2008.22.861 +[Abstract](197) +[PDF](346.6KB)
Abstract:
We consider an integrodifferential reaction-diffusion system on a multidimensional spatial domain, subject to homogeneous Neumann boundary conditions. This system finds applications in population dynamics and it is characterized by nonlocal delay terms depending on both the temporal and the spatial variables. The distributed time delay effects are represented by memory kernels which decay exponentially but they are not necessarily monotonically decreasing. We first show how to construct a (dissipative) dynamical system on a suitable phase-space. Then we discuss the existence of the global attractor as well as of an exponential attractor.
Positive solutions of an integro-differential equation in all space with singular nonlinear term
Giuseppe Maria Coclite and Mario Michele Coclite
2008, 22(4): 885-907 doi: 10.3934/dcds.2008.22.885 +[Abstract](302) +[PDF](266.4KB)
Abstract:
We prove the existence of a positive solution in $W_{loc}^{2,q}$ for a semilinear elliptic integro-differential problem in $\mathbb{R}^N.$ The integral operator of the equation depends on a nonlinear function that is singular in the origin. Moreover, we prove that the averages of the solution and its gradient on the balls $\{x\in\mathbb{R}^N; |x| \le R\}, R>0,$ vanish as $R\to \infty.$
Uniform attractors for a phase transition model coupling momentum balance and phase dynamics
Pierluigi Colli and Antonio Segatti
2008, 22(4): 909-932 doi: 10.3934/dcds.2008.22.909 +[Abstract](271) +[PDF](328.1KB)
Abstract:
In this paper we are concerned with the uniform attractor for a nonautonomous dynamical system related to the Frémond thermo-mechanical model of shape memory alloys. The dynamical system consists of a diffusive equation for the phase proportions coupled with the hyperbolic momentum balance equation, in the case when a damping term is considered in the latter and the temperature field is prescribed. We prove that the solution to the related initial-boundary value problem yields a semiprocess which is continuous on the proper phase space and satisfies a dissipativity property. Then we show the existence of a unique compact and connected uniform attractor for the system.
On a class of elliptic and parabolic equations in convex domains without boundary conditions
Giuseppe Da Prato and Alessandra Lunardi
2008, 22(4): 933-953 doi: 10.3934/dcds.2008.22.933 +[Abstract](206) +[PDF](247.1KB)
Abstract:
We consider the operator $\A u = \frac{1}{2} \Delta u - \langle DU, Du\right$, where $U $ is a convex real function defined in a convex open set $\O \subset \R^N$ and $\lim_{|x|\to \infty} U(x) = \lim_{ x \to \partial \O} U(x)$ $ =$ $ +\infty$. We study the realization of $\A $ in the spaces $C_{b}(\overline{\O})$, $C_{b}(\O)$ and $B_{b}(\O)$, and prove several properties of the associated Markov semigroup. In contrast with the case of bounded coefficients, elliptic equations and parabolic Cauchy problems such as (3) and (4) below are uniquely solvable in reasonable classes of functions, without imposing any boundary condition. We prove that the associated semigroup coincides with the transition semigroup of a stochastic variational inequality on $C_{b}(\overline{\O})$.
A functional calculus approach for the rational approximation with nonuniform partitions
Hassan Emamirad and Arnaud Rougirel
2008, 22(4): 955-972 doi: 10.3934/dcds.2008.22.955 +[Abstract](200) +[PDF](229.9KB)
Abstract:
By introducing $M$-functional calculus we generalize some results of Brenner and Thomée on the stability and convergence of rational approximation schemes of bounded semigroups for nonuniform time steps. We give also the rate of convergence for the approximation of the time derivative of these semigroups.
Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces
Angelo Favini, Rabah Labbas, Stéphane Maingot, Hiroki Tanabe and Atsushi Yagi
2008, 22(4): 973-987 doi: 10.3934/dcds.2008.22.973 +[Abstract](197) +[PDF](193.8KB)
Abstract:
In this paper we give new results on complete abstract second order differential equations of elliptic type in the framework of Hölder spaces, extending those given in [4] and [5]. More precisely we study $u^{\prime \prime }+2Bu^{\prime }+Au=f$ in the case when $f$ is Hölder continuous and under some natural assumptions on the operators $A$ and $B$. We give necessary and sufficient conditions of compatibility to obtain a strict solution $u$ and also to ensure that the strict solution has the maximal regularity property.
An $L^p$-approach to singular linear parabolic equations with lower order terms
Angelo Favini, Alfredo Lorenzi, Hiroki Tanabe and Atsushi Yagi
2008, 22(4): 989-1008 doi: 10.3934/dcds.2008.22.989 +[Abstract](239) +[PDF](225.9KB)
Abstract:
Singular means here that the parabolic equation is neither in normal form nor can it be reduced to such a form. For this class of problems we generalizes the results proved in [4] introducing first-order terms.
The non-isothermal Allen-Cahn equation with dynamic boundary conditions
Ciprian G. Gal and Maurizio Grasselli
2008, 22(4): 1009-1040 doi: 10.3934/dcds.2008.22.1009 +[Abstract](273) +[PDF](356.3KB)
Abstract:
We consider a model of nonisothermal phase transitions taking place in a bounded spatial region. The order parameter $\psi$ is governed by an Allen-Cahn type equation which is coupled with the equation for the temperature $\theta$. The former is subject to a dynamic boundary condition recently proposed by some physicists to account for interactions with the walls. The latter is endowed with a boundary condition which can be a standard one (Dirichlet, Neumann or Robin) or a dynamic one of Wentzell type. We thus formulate a class of initial and boundary value problems whose local existence and uniqueness is proven by means of a fixed point argument. The local solution becomes global owing to suitable a priori estimates. Then we analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. In particular, we demonstrate the existence of the global attractor as well as of an exponential attractor.
Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation
Ciprian G. Gal and Hao Wu
2008, 22(4): 1041-1063 doi: 10.3934/dcds.2008.22.1041 +[Abstract](225) +[PDF](328.3KB)
Abstract:
In this article, we consider a Cahn-Hilliard model with boundary conditions of Wentzell type and mass conservation. We show that each solution of this problem converges to a steady state as time goes to infinity, provided that the potential function $f$ is real analytic and satisfies certain growth assumptions. Estimates of the rate of convergence to equilibrium are obtained as well. We also recall some results about the existence of global and exponential attractors and their properties.
Stable discretizations of the Cahn-Hilliard-Gurtin equations
Sami Injrou and Morgan Pierre
2008, 22(4): 1065-1080 doi: 10.3934/dcds.2008.22.1065 +[Abstract](227) +[PDF](273.2KB)
Abstract:
We study space and time discretizations of the Cahn-Hilliard-Gurtin equations with a polynomial nonlinearity. We first consider a space semi-discrete version of the equations, and we prove in particular that any solution converges to a steady state (as in the continuous case). Then, we study the numerical stability of the fully discrete scheme obtained by applying the Euler backward scheme to the space semi-discrete problem. In particular, we show that this fully discrete problem is unconditionally stable. Numerical simulations in one space dimension conclude the paper.
Quasi-$m$-accretivity of Schrödinger operators with singular first-order coefficients
Noboru Okazawa and Tomomi Yokota
2008, 22(4): 1081-1090 doi: 10.3934/dcds.2008.22.1081 +[Abstract](197) +[PDF](174.8KB)
Abstract:
The Schrödinger operator $T = (i\nabla +b)^2+a \cdot \nabla + q$ on $\mathbb{R}^N$ is considered for $N \ge 2$. Here $a=(a_{j})$ and $b=(b_{j})$ are real-vector-valued functions on $\mathbb{R}^N$, while $q$ is a complex-scalar-valued function on $\mathbb{R}^N$. Over twenty years ago late Professor Kato proved that the minimal realization $T_{min}$ is essentially quasi-$m$-accretive in $L^2(\mathbb{R}^N)$ if, among others, $(1+|x|)^{-1}a_j \in L^4(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$. In this paper it is shown that under some additional conditions the same conclusion remains true even if $a_j \in L^4_{loc}(\mathbb{R}^N)$.
Exponential attractors for competing species model with cross-diffusions
Atsushi Yagi
2008, 22(4): 1091-1120 doi: 10.3934/dcds.2008.22.1091 +[Abstract](240) +[PDF](344.0KB)
Abstract:
This paper is concerned with the competing species model presented by Shigesada-Kawasaki-Teramoto in 1979. Under a suitable condition on self-diffusions and cross-diffusions, we construct a dynamical system determined from the model. Furthermore, under the same condition we construct exponential attractors of the dynamical system.

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