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Discrete & Continuous Dynamical Systems - S

2009 , Volume 2 , Issue 3

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Preface
Irena Lasiecka, Mitsuhiro Nakao and Grozdena Todorova
2009, 2(3): i-ii doi: 10.3934/dcdss.2009.2.3i +[Abstract](258) +[PDF](28.4KB)
Abstract:
This issue comprises a selection of papers in the general area of analysis and control of systems described by non-linear evolutionary equations, that are relevant to applications in mathematical physics. Models considered range from classical non-linear wave and heat equations to quite complex systems consisting of two or more coupled equations. In this latter case, coupling often occurs between two different types of dynamics - say, a hyperbolic component and a parabolic component - with coupling in various forms, throughout the interior of the spatial domain and/or at the interface between the two media. Illustrations include systems of non-linear thermo-elasticity; fluid structure- and acoustic-structure interactions; electro-magnetism among others.
    Dynamical models such as these are frequently encountered in modern technological applications. In recent years, they have attracted considerable attention and many new results and developments have become available.
    Papers collected in this volume address and present some of these advances, with particular emphasis on newly developed techniques that bear on further progress in the field.

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Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction
George Avalos and Roberto Triggiani
2009, 2(3): 417-447 doi: 10.3934/dcdss.2009.2.417 +[Abstract](540) +[PDF](343.2KB)
Abstract:
This paper is focused on an established, genuinely physical fluid-structure interaction model, herein the structure is immersed in a fluid with coupling taking place at the boundary interface between the two media. Mathematically, the model is a coupled parabolic-hyperbolic system of two partial differential equations in 3-d with nonstandard coupling at the boundary interface. Fluid and structure are mathematically expressed by the (dynamic) Stokes system (parabolic) and the Lamé system (hyperbolic), respectively. The main claim presented is a contraction semigroup well-posedness result on the natural space of finite energy. There are two main features in the analysis: (i) a nonstandard elimination of the pressure term, as the boundary coupling between fluid and structure rules out application of the classical Leray/Helmoltz projection; (ii) a nonstandard usage of the Babuška-Brezzi "inf-sup" theory to assert maximal dissipativity of the candidate generator. A unified treatment includes both undamped and (perhaps, partially) damped boundary conditions at the interface. With the generator explicitly at hand, an analysis of its point spectrum on the imaginary axis is also included. In the undamped case, it depends on the geometry of the structure. In the case of full boundary damping, it implies as a by-product a strong stability result for the solutions by soft methods.
Singularly non-autonomous semilinear parabolic problems with critical exponents
Alexandre Nolasco de Carvalho and Marcelo J. D. Nascimento
2009, 2(3): 449-471 doi: 10.3934/dcdss.2009.2.449 +[Abstract](230) +[PDF](289.0KB)
Abstract:
In this work we consider initial value problems of the form

$\frac{dx}{dt} + A(t)x = f(t,x)$
$x(\tau)=x_0,$

in a Banach space $X$ where $A(t):D\subset X\to X$ is a linear, closed and unbounded operator which is sectorial for each $t$. We show local well posedness for the case when the nonlinearity $f$ grows critically. Applications to semilinear parabolic equations and strongly damped wave equations are considered.

On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition
M. Eller
2009, 2(3): 473-481 doi: 10.3934/dcdss.2009.2.473 +[Abstract](322) +[PDF](154.0KB)
Abstract:
The dynamic Maxwell equations with a conservative boundary condition are considered. A boundary regularity result for classical solutions is proved. This result is remarkable since the boundary condition does not satisfy the uniform Lopatinskii (Kreiss-Sakamoto) condition.
Fundamental solutions for wave equation in Robertson-Walker model of universe and $L^p-L^q$ -decay estimates
Karen Yagdjian and Anahit Galstian
2009, 2(3): 483-502 doi: 10.3934/dcdss.2009.2.483 +[Abstract](280) +[PDF](262.2KB)
Abstract:
We use constructed in [24] the fundamental solutions of the wave equation arising in the Robertson-Walker model of universe to derive the $L^p-L^q$-decay estimates for the solutions of the equation with and without a source term.
Global existence and exponential decay rates for the Westervelt equation
Barbara Kaltenbacher and Irena Lasiecka
2009, 2(3): 503-523 doi: 10.3934/dcdss.2009.2.503 +[Abstract](545) +[PDF](239.8KB)
Abstract:
We consider the Westervelt equation which models propagation of sound in a fluid medium. This is an accepted in nonlinear acoustics model which finds a multitude of applications in medical imaging and therapy. The PDE model consists of the second order in time evolution which is both quasilinear and degenerate. Degeneracy depends on the fluctuations of the acoustic pressure.
    Our main results are : (1) global well-posedness, (2) exponential decay rates for the energy function corresponding to both weak and strong solutions. The proof is based on (i) application of a suitable fixed point theorem applied to an appropriate formulation of the PDE which exhibits analyticity properties of the underlying linearised semigroup, (ii) exploitation of decay rates associated with the dissipative mechanism along with barrier's method leading to global wellposedness. The obtained result holds for all times, provided that the initial data are taken from a suitably small ball characterized by the parameters of the equation.
Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators (II)
Antonio Avantaggiati and Paola Loreti
2009, 2(3): 525-545 doi: 10.3934/dcdss.2009.2.525 +[Abstract](336) +[PDF](241.7KB)
Abstract:
In this paper we study Hopf-Lax formulas, hypercontractivity, ultracontractivity, logarithmic Sobolev inequalities for a class of first order Hamilton-Jacobi equations.
Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation
Cleverson R. da Luz and Gustavo Alberto Perla Menzala
2009, 2(3): 547-558 doi: 10.3934/dcdss.2009.2.547 +[Abstract](393) +[PDF](173.8KB)
Abstract:
We consider the Maxwell system with variable anisotropic coefficients in a bounded domain $\Omega$ of $\mathbb{R}^3$. The boundary conditions are of Silver-Muller's type. We proved that the total energy decays exponentially fast to zero as time approaches infinity. This result is well known in the case of isotropic coefficients. We make use of modified multipliers with the help of an elliptic problem and some technical assumptions on the permittivity and permeability matrices.
Stability of the heat and of the wave equations with boundary time-varying delays
Serge Nicaise, Julie Valein and Emilia Fridman
2009, 2(3): 559-581 doi: 10.3934/dcdss.2009.2.559 +[Abstract](540) +[PDF](248.1KB)
Abstract:
Exponential stability analysis via Lyapunov method is extended to the one-dimensional heat and wave equations with time-varying delay in the boundary conditions. The delay function is admitted to be time-varying with an a priori given upper bound on its derivative, which is less than $1$. Sufficient and explicit conditions are derived that guarantee the exponential stability. Moreover the decay rate can be explicitly computed if the data are given.
On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms
Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha and Daniel Toundykov
2009, 2(3): 583-608 doi: 10.3934/dcdss.2009.2.583 +[Abstract](599) +[PDF](327.1KB)
Abstract:
This paper is concerned with the study of the nonlinearly damped system of wave equations with Dirichlét boundary conditions:

$u_{t t}$ $- \Delta u + |u_t|^{m-1}u_t = F_u(u,v) \text{ in }\Omega\times ( 0,\infty )$,
$v_{t t}$$ - \Delta v + |v_t|^{r-1}v_t = F_v(u,v) \text{ in }\Omega\times( 0,\infty )$,

where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n=1,2,3$ with a smooth boundary $\partial\Omega=\Gamma$ and $F$ is a $C^1$ function given by

$ F(u,v)=\alpha|u+v|^{p+1}+ 2\beta |uv|^{\frac{p+1}{2}}. $

Under some conditions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain several results on the global existence, uniform decay rates, and blow up of solutions in finite time when the initial energy is nonnegative.

Higher order energy decay rates for damped wave equations with variable coefficients
Petronela Radu, Grozdena Todorova and Borislav Yordanov
2009, 2(3): 609-629 doi: 10.3934/dcdss.2009.2.609 +[Abstract](288) +[PDF](244.0KB)
Abstract:
Under appropriate assumptions the energy of wave equations with damping and variable coefficients $c(x)$$u_{t t}$-div$(b(x)\nabla u)+a(x)u_t =h(x,t)$ has been shown to decay. Determining the decay rate for the higher order energies of the $k$th order spatial and time derivatives has been an open problem with the exception of some sparse results obtained for $k=1,2$. We establish the sharp gain in the decay rate for all higher order energies in terms of the first energy, and also obtain the sharp gain of decay rates for the $L^2$ norms of the higher order spatial derivatives. The results concern weighted (in time) and also pointwise (in time) energy decay estimates. We also obtain $L^\infty$ estimates for the solution $u$ in dimension $n=3$. As an application we compute explicit decay rates for all energies which involve the dimension $n$ and the bounds for the coefficients $a(x)$ and $b(x)$ in the case $c (x)=1$ and $h(x,t)=0.$
Interface conditions for a singular reaction-diffusion system
Thomas I. Seidman
2009, 2(3): 631-643 doi: 10.3934/dcdss.2009.2.631 +[Abstract](254) +[PDF](420.5KB)
Abstract:
For a chemical reaction/diffusion system, a very fast reaction $A+B\to C$ implies non-coexistence of $A,B$ with resulting interfaces. We try to understand how these interfaces evolve in time. In particular, we seek a characterizing system of equations and conditions for the sharp interface limit: when this fast reaction is taken as infinitely fast.
Unique continuation of boundary over-determined Stokes and Oseen eigenproblems
Roberto Triggiani
2009, 2(3): 645-677 doi: 10.3934/dcdss.2009.2.645 +[Abstract](235) +[PDF](381.8KB)
Abstract:
We provide several radically different proofs of the following unique continuation result: The Oseen eigenvalue problem with over-determined homogeneous Cauchy data for the velocity field on the boundary implies the zero solution, at least if the equilibrium solution is sufficiently 'small.' In particular, this unique continuation result from over-determined boundary data holds true for the Stokes problem.
The semilinear Klein-Gordon equation in de Sitter spacetime
Karen Yagdjian
2009, 2(3): 679-696 doi: 10.3934/dcdss.2009.2.679 +[Abstract](314) +[PDF](223.3KB)
Abstract:
In this article we study the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation $\square_g$ $\phi-m^2 \phi = -|\phi |^p $ with the small mass $m \le n/2$ in de Sitter spacetime with the metric $g$. We prove that for every $p>1$ large energy solutions blow up, while for the small energy solutions we give a borderline $p=p(m,n)$ for the global in time existence. The consideration is based on the representation formulas for the solution of the Cauchy problem and on some generalizations of Kato's lemma.
On shallow shell equations
Peng-Fei Yao
2009, 2(3): 697-722 doi: 10.3934/dcdss.2009.2.697 +[Abstract](284) +[PDF](305.1KB)
Abstract:
We present explicit formulas for the shallow shell model consisting of a couple of a wave equation and a plate equation, where the middle surface is viewed as a natural manifold with the induced metric from the classical Euclidean space of three dimensions. The Green formula for the shallow is given by the displacement field which expresses the relationship between the interior and the boundary. Next, the ellipticity of the strain energy for the shallow shell is studied under some curvature assumptions on the middle surface. Finally, the motion equations for shallow shells are obtained in terms of the displacement field as an unknown. The new ingredients in these formulas are that they take a form which is not described by a coordinate patch to provide the shell theory with the modern geometry.

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