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## Selected Abstracts## Fast 3D 1H MRSI of the corticospinal tract in pediatric brain JOURNAL OF MAGNETIC RESONANCE IMAGING, Issue 1 2009Dong-Hyun Kim PhDAbstract Purpose To develop a 1H magnetic resonance spectroscopic imaging (MRSI) sequence that can be used to image infants/children at 3T and by combining it with diffusion tensor imaging (DTI) tractography, extract relevant metabolic information corresponding to the corticospinal tract (CST). Materials and Methods A fast 3D MRSI sequence was developed for pediatric neuroimaging at 3T using spiral k-space readout and dual band RF pulses (32 × 32 × 8 cm field of view [FOV], 1 cc iso-resolution, TR/TE = 1500/130, 6:24 minute scan). Using DTI tractography to identify the motor tracts, spectra were extracted from the CSTs and quantified. Initial data from infants/children with suspected motor delay (n = 5) and age-matched controls (n = 3) were collected and N -acetylaspartate (NAA) ratios were quantified. Results The average signal-to-noise ratio of the NAA peak from the studies was ,22. Metabolite profiles were successfully acquired from the CST by using DTI tractography. Decreased NAA ratios in those with motor delay compared to controls of ,10% at the CST were observed. Conclusion A fast and robust 3D MRSI technique targeted for pediatric neuroimaging has been developed. By combining with DTI tractography, metabolic information from the CSTs can be retrieved and estimated. By combining DTI and 3D MRSI, spectral information from various tracts can be obtained and processed. J. Magn. Reson. Imaging 2009;29:1,6. © 2008 Wiley-Liss, Inc. [source] ## Singular integral operator, Hardy,Morrey space estimates for multilinear operators and Navier,Stokes equations MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2010Henggeng WangAbstract After establishing the molecule characterization of the Hardy,Morrey space, we prove the boundedness of the singular integral operator and the Riesz potential. We also obtain the Hardy,Morrey space estimates for multilinear operators satisfying certain vanishing moments. As an application, we study the existence and the uniqueness of the solutions to the Navier,Stokes equations for the initial data in the Hardy,Morrey space ,,(p,n) for q as small as possible. Here, the Hardy,Morrey space estimates for multilinear operators are important tools. Copyright © 2010 John Wiley & Sons, Ltd. [source] ## Global well-posedness of the Cauchy problem for certain magnetohydrodynamic-, models MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2010Yi DuAbstract This paper is devoted to study the Cauchy problem for certain incompressible magnetohydrodynamics-, model. In the Sobolev space with fractional index s>1, we proved the local solutions for any initial data, and global solutions for small initial data. Furthermore, we also prove that as ,,0, the MHD-, model reduces to the MHD equations, and the solutions of the MHD-, model converge to a pair of solutions for the MHD equations. Copyright © 2010 John Wiley & Sons, Ltd. [source] ## Existence of front solutions for a nonlocal transport problem describing gas ionization MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2010M. GüntherAbstract We discuss a moving boundary problem arising from a model of gas ionization in the case of negligible electron diffusion and suitable initial data. It describes the time evolution of an ionization front. Mathematically, it can be considered as a system of transport equations with different characteristics for positive and negative charge densities. We show that only advancing fronts are possible and prove short-time well posedness of the problem in Hölder spaces of functions. Technically, the proof is based on a fixed-point argument for a Volterra-type system of integral equations involving potential operators. It crucially relies on estimates of such operators with respect to variable domains in weighted Hölder spaces and related calculus estimates. Copyright © 2010 John Wiley & Sons, Ltd. [source] ## Localization for a doubly degenerate parabolic equation with strongly nonlinear sources MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2010Zhaoyin XiangAbstract In this paper, we study the strict localization for the doubly degenerate parabolic equation with strongly nonlinear sources, We prove that, for non-negative compactly supported initial data, the strict localization occurs if and only if q,m(p,1). Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Global regular solutions to Cahn,Hilliard system coupled with viscoelasticity MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2009Irena PawAbstract In this paper we prove the existence and uniqueness of a global in time, regular solution to the Cahn,Hilliard system coupled with viscoelasticity. The system arises as a model, regularized by a viscous damping, of phase separation process in a binary deformable alloy quenched below a critical temperature. The key tools in the analysis are estimates of absorbing type with the property of exponentially time-decreasing contribution of the initial data. Such estimates allow not only to prolong the solution step by step on the infinite time interval but also to conclude the existence of an absorbing set. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## A weak solution approach to a reaction,diffusion system modeling pattern formation on seashells MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2009Jan KelkelAbstract We investigate a reaction,diffusion system proposed by H. Meinhardt as a model for pattern formation on seashells. We give a new proof for the existence of a local weak solution for general initial conditions and parameters upon using an iterative approach. Furthermore, the solution is shown to exist globally for suitable initial data. The behavior of the solution in time and space is illustrated through numerical simulations. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2009Aissa GuesmiaAbstract In this paper we consider the following Timoshenko system: with Dirichlet boundary conditions and initial data where a, b, g and h are specific functions and ,1, ,2, k1, k2 and L are given positive constants. We establish a general stability estimate using the multiplier method and some properties of convex functions. Without imposing any growth condition on h at the origin, we show that the energy of the system is bounded above by a quantity, depending on g and h, which tends to zero as time goes to infinity. Our estimate allows us to consider a large class of functions h with general growth at the origin. We use some examples (known in the case of wave equations and Maxwell system) to show how to derive from our general estimate the polynomial, exponential or logarithmic decay. The results of this paper improve and generalize some existing results in the literature and generate some interesting open problems. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Blow up for a Cauchy viscoelastic problem with a nonlinear dissipation of cubic convolution type MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2009Shengqi YuAbstract In this paper, we consider a Cauchy viscoelastic problem with a nonlinear source of polynomial type and a nonlinear dissipation of cubic convolution type involving a singular kernel. Under suitable conditions on the initial data and the relaxation functions, it is proved that the solution of this particular problem blows up in finite time. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Point-wise decay estimate for the global classical solutions to quasilinear hyperbolic systems MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2009Yi ZhouAbstract In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19:1263,1317; Nonlinear Anal. 1997; 28:1299,1322; Chin. Ann. Math. 2004; 25B:37,56). We give a new, very simple proof of this result and also give a sharp point-wise decay estimate of the solution. Then, we consider the mixed initial-boundary-value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12(1):59,78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point-wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Polynomial and analytic stabilization of a wave equation coupled with an Euler,Bernoulli beam MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2009Kaïs AmmariAbstract We consider a stabilization problem for a model arising in the control of noise. We prove that in the case where the control zone does not satisfy the geometric control condition, B.L.R. (see Bardos et al. SIAM J. Control Optim. 1992; 30:1024,1065), we have a polynomial stability result for all regular initial data. Moreover, we give a precise estimate on the analyticity of reachable functions where we have an exponential stability. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## On the well-posedness of the Cauchy problem for an MHD system in Besov spaces MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2009Changxing MiaoAbstract This paper is devoted to the study of the Cauchy problem of incompressible magneto-hydrodynamics system in the framework of Besov spaces. In the case of spatial dimension n,3, we establish the global well-posedness of the Cauchy problem of an incompressible magneto-hydrodynamics system for small data and the local one for large data in the Besov space , (,n), 1,p<, and 1,r,,. Meanwhile, we also prove the weak,strong uniqueness of solutions with data in , (,n),L2(,n) for n/2p+2/r>1. In the case of n=2, we establish the global well-posedness of solutions for large initial data in homogeneous Besov space , (,2) for 2 ## Boundedness and exponential stabilization in a signal transduction model with diffusion MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2008Michael WinklerAbstract The influence of diffusion in a model arising in the description of signal transduction pathways in living cells is investigated. It is proved that all solutions of the corresponding semilinear parabolic system, consisting of four equations, are global in time and bounded. Under the additional assumption that certain two of the diffusion coefficients are equal, it is furthermore demonstrated that all solutions approach a spatially homogeneous steady state as t,,,,. This equilibrium is uniquely determined by the initial data, and the rate of convergence is shown to be at least exponential. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Stability of weak solutions to the compressible Navier,Stokes equations in bounded annular domains MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2008Jishan FanAbstract We prove the Lipschitz continuous dependence on initial data of global spherically symmetric weak solutions to the Navier,Stokes equations of a viscous polytropic ideal gas in bounded annular domains with the initial data in the Lebesgue spaces. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Well-posedness, smooth dependence and centre manifold reduction for a semilinear hyperbolic system from laser dynamics MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2007Mark LichtnerAbstract We prove existence, uniqueness, regularity and smooth dependence of the weak solution on the initial data for a semilinear, first order, dissipative hyperbolic system with discontinuous coefficients. Such hyperbolic systems have successfully been used to model the dynamics of distributed feedback multisection semiconductor lasers. We show that in a function space of continuous functions the weak solutions generate a smooth skew product semiflow. Using slow fast structure and dissipativity we prove the existence of smooth exponentially attracting invariant centre manifolds for the non-autonomous model. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Convergence rates toward the travelling waves for a model system of the radiating gas MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2007Masataka NishikawaAbstract The present paper is concerned with an asymptotics of a solution to the model system of radiating gas. The previous researches have shown that the solution converges to a travelling wave with a rate (1 + t),1/4 as time t tends to infinity provided that an initial data is given by a small perturbation from the travelling wave in the suitable Sobolev space and the perturbation is integrable. In this paper, we make more elaborate analysis under suitable assumptions on initial data in order to obtain shaper convergence rates than previous researches. The first result is that if the initial data decays at the spatial asymptotic point with a certain algebraic rate, then this rate reflects the time asymptotic convergence rate. Precisely, this convergence rate is completely same as the spatial convergence rate of the initial perturbation. The second result is that if the initial data is given by the Riemann data, an admissible weak solution, which has a discontinuity, converges to the travelling wave exponentially fast. Both of two results are proved by obtaining decay estimates in time through energy methods with suitably chosen weight functions. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Global existence and uniform stability of solutions for a quasilinear viscoelastic problem MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2007Salim A. MessaoudiAbstract In this paper the nonlinear viscoelastic wave equation in canonical form with Dirichlet boundary condition is considered. By introducing a new functional and using the potential well method, we show that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Local energy decay for linear wave equations with non-compactly supported initial data MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2004Ryo IkehataAbstract A local energy decay problem is studied to a typical linear wave equation in an exterior domain. For this purpose, we do not assume any compactness of the support on the initial data. This generalizes a previous famous result due to Morawetz (Comm. Pure Appl. Math. 1961; 14:561,568). In order to prove local energy decay we mainly apply two types of new ideas due to Ikehata,Matsuyama (Sci. Math. Japon. 2002; 55:33,42) and Todorova,Yordanov (J. Differential Equations 2001; 174:464). Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2004Taeko YamazakiAbstract We consider the unique global solvability of initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole Euclidean space for dimension larger than three. The following sufficient condition is known: initial data is sufficiently small in some weighted Sobolev spaces for the whole space case; the generalized Fourier transform of the initial data is sufficiently small in some weighted Sobolev spaces for the exterior domain case. The purpose of this paper is to give sufficient conditions on the usual Sobolev norm of the initial data, by showing that the global solvability for this equation follows from a time decay estimate of the solution of the linear wave equation. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Evolution completeness of separable solutions of non-linear diffusion equations in bounded domains MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2004V. A. GalaktionovAbstract As a basic example, we consider the porous medium equation (m > 1) (1) where , , ,N is a bounded domain with the smooth boundary ,,, and initial data . It is well-known from the 1970s that the PME admits separable solutions , where each ,k , 0 satisfies a non-linear elliptic equation . Existence of at least a countable subset , = {,k} of such non-linear eigenfunctions follows from the Lusternik,Schnirel'man variational theory from the 1930s. The first similarity pattern t,1/(m,1),0(x), where ,0 > 0 in ,, is known to be asymptotically stable as t , , and attracts all nontrivial solutions with u0 , 0 (Aronson and Peletier, 1981). We show that if , is discrete, then it is evolutionary complete, i.e. describes the asymptotics of arbitrary global solutions of the PME. For m = 1 (the heat equation), the evolution completeness follows from the completeness-closure of the orthonormal subset , = {,k} of eigenfunctions of the Laplacian , in L2. The analysis applies to the perturbed PME and to the p -Laplacian equations of second and higher order. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## The asymptotic behaviour of global smooth solutions to the multi-dimensional hydrodynamic model for semiconductors MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2003Ling HsiaoAbstract We establish the global existence of smooth solutions to the Cauchy problem for the multi-dimensional hydrodynamic model for semiconductors, provided that the initial data are perturbations of a given stationary solutions, and prove that the resulting evolutionary solution converges asymptotically in time to the stationary solution exponentially fast. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Initial boundary value problem for a class of non-linear strongly damped wave equations MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2003Yang ZhijianThe paper studies the existence, asymptotic behaviour and stability of global solutions to the initial boundary value problem for a class of strongly damped non-linear wave equations. By a H00.5ptk-Galerkin approximation scheme, it proves that the above-mentioned problem admits a unique classical solution depending continuously on initial data and decaying to zero as t,+,as long as the non-linear terms are sufficiently smooth; they, as well as their derivatives or partial derivatives, are of polynomial growth order and the initial energy is properly small. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Initial boundary value problem for the evolution system of MHD type describing geophysical flow in three-dimensional domains MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2003Chunshan ZhaoAbstract The initial boundary value problem for the evolution system describing geophysical flow in three-dimensional domains was considered. The existence and uniqueness of global strong solution to the evolution system were proved under assumption on smallness of data. Moreover, solvable compatibility conditions of initial data and boundary values which guarantee the existence and uniqueness of global strong solution were discussed. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Global existence for the Vlasov,Darwin system in ,3 for small initial data MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2003Saïd BenachourWe prove the global existence of weak solutions to the Vlasov,Darwin system in R3 for small initial data. The Vlasov,Darwin system is an approximation of the Vlasov,Maxwell model which is valid when the characteristic speed of the particles is smaller than the light velocity, but not too small. In contrast to the Vlasov,Maxwell system, the total energy conservation does not provide an L2-bound on the transverse part of the electric field. This difficulty may be overcome by exploiting the underlying elliptic structure of the Darwin equations under a smallness assumption on the initial data. We finally investigate the convergence of the Vlasov,Darwin system towards the Vlasov,Poisson system. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Complex-distance potential theory, wave equations, and physical wavelets MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16-18 2002Gerald KaiserPotential theory in ,n is extended to ,n by analytically continuing the Euclidean distance function. The extended Newtonian potential ,(z) is generated by a (non-holomorphic) source distribution ,,(z) extending the usual point source ,(x). With Minkowski space ,n, 1 embedded in ,n+1, the Laplacian ,n+1 restricts to the wave operator ,n,1 in ,n, 1. We show that ,,(z) acts as a propagator generating solutions of the wave equation from their initial values, where the Cauchy data need not be assumed analytic. This generalizes an old result by Garabedian, who established a connection between solutions of the boundary-value problem for ,n+1 and the initial-value problem for ,n,1 provided the boundary data extends holomorphically to the initial data. We relate these results to the physical avelets introduced previously. In the context of Clifford analysis, our methods can be used to extend the Borel,Pompeiu formula from ,n+1 to ,n+1, where its riction to Minkowski space ,n, 1 provides solutions for time-dependent Maxwell and Dirac equations. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Evolution free boundary problem for equations of viscous compressible heat-conducting capillary fluids MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2001Ewa ZadrzyIn the paper the global motion of a viscous compressible heat conducting capillary fluid in a domain bounded by a free surface is considered. Assuming that the initial data are sufficiently close to a constant state and the external force vanishes we prove the existence of a global-in-time solution which is close to the constant state for any moment of time. The solution is obtained in such Sobolev,Slobodetskii spaces that the velocity, the temperature and the density of the fluid have $W_2^{2+\alpha,1+\alpha/2}$\nopagenumbers\end, $W_2^{2+\alpha,1+\alpha/2}$\nopagenumbers\end and $W_2^{1+\alpha,1/2+\alpha/2}$\nopagenumbers\end ,regularity with ,,(¾, 1), respectively. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Oscillatory asymptotic expansion for semilinear wave equation MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2001Stefania Di PomponioWe study oscillatory properties of the solution to semilinear wave equation, assuming oscillatory terms in initial data have sufficiently small amplitude. The main result gives an a priori estimate of the remainder in the approximation by means of the method of geometric optics. The method of establishing this estimate is based on a combination between energy type estimates for transport equation and Sobolev embedding. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Compressible Navier,Stokes system in 1-D MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2001Piotr Bogus, aw MuchaAbstract The compressible barotropic Navier,Stokes system in monodimensional case with a Neumann boundary condition given on a free boundary is considered. The global existence with uniformly boundedness for large initial data and a positive force is proved. The result concerning an asymptotic behavior shows that the solutions tends to the stationary solution. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Local energy decay for a class of hyperbolic equations with constant coefficients near infinity MATHEMATISCHE NACHRICHTEN, Issue 5 2010Shintaro AikawaAbstract A uniform local energy decay result is derived to a compactly perturbed hyperbolic equation with spatial vari¬able coefficients. We shall deal with this equation in an N -dimensional exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data and the equation includes anisotropic variable coefficients {ai(x): i = 1, 2, ,, N }, which are not necessarily equal to each other (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Minimising the variance under convex constraints PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003Ulrich Hirth Dr. rer. nat.I treat the problem of minimising the variance functional on a certain closed convex subset of L2(P) by means of the variational inequality, obtaining an explicit formula for the solution and analysing the dependence of the solution on the initial data. [source] |