The Cauchy problem of the Ward equation

We generalize the results of [J. Villarroel, The inverse problem for Ward's system, Stud. Appl. Math. 83 (1990) 211-222; A.S. Fokas, T.A. Ioannidou, The inverse spectral theory for the Ward equation and for the 2+1 chiral model, Comm. Appl. Anal. 5 (2001) 235-246; B. Dai, C.L. Terng, K. Uhlenbeck, On the space-time Monopole equation, arXiv:math.DG/0602607] to study the inverse scattering problem of the Ward equation with non-small data and solve the Cauchy problem of the Ward equation with a non-small purely continuous scattering data.     

The spectrum of the Leray transform for convex Reinhardt domains in

The Leray transform and related boundary operators are studied for a class of convex Reinhardt domains in . Our class is self-dual; it contains some domains with less than C²-smooth boundary and also some domains with smooth boundary and degenerate Levi form. L²-regularity is proved, and essential spectra are computed with respect to a family of boundary measures which includes surface measure. A duality principle is established providing explicit unitary equivalence between operators on domains in our class and operators on the corresponding polar domains. Many of these results are new even for the classical case of smoothly bounded strongly convex Reinhardt domains.     

Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation

We consider the existence, both locally and globally in time, and the blow-up of solutions for the Cauchy problem of the generalized damped multidimensional Boussinesq equation.     

On the Cauchy problem for a generalized Boussinesq equation

In this paper, we consider the Cauchy problem for a generalized Boussinesq equation. We show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem.     

Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations

In this paper we consider the Cauchy problem of multidimensional generalized double dispersion equations utt−Δu−Δutt+Δ²u=Δf(u), where f(u)=ap|u|. By potential well method we prove the existence and nonexistence of global weak solution without establishing the local existence theory. And we derive some sharp conditions for global existence and lack of global existence solution.     

On the uniform decay in viscoelastic problems in

In this paper we consider a linear Cauchy viscoelastic problem. We show that, for compactly supported initial data and for an exponentially decaying relaxation function, the decay of the first energy of solution is polynomial. The finite-speed propagation is used to compensate for the lack of Poincaré’s inequality in .     

Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

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.     

A fixed point result in Menger spaces using a real function

The main result of this paper is a fixed point theorem of self-mappings in Menger spaces which satisfy certain inequality. This inequality involves a class of real functions which we call Φ-functions. As a corollary we obtain a result in the corresponding metric spaces. The result is supported by an example. The class of real functions we have used is the conceptual extension of altering distance functions used in metric fixed point theory.      

A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity

In one space dimension, a non-local elastic model is based ona single integral law, giving the stress when the strain isknown at all spatial points. In this study, we first derivea higher-order Boussinesq equation using locally non-lineartheory of 1D non-local elasticity and then we are able to showthat under certain conditions the Cauchy problem is globallywell-posed.     

Important physical parameters of Bi2O3 thin films found by applying several models for optical data

Different optical parameters for thin solid films can be computed as functions of wavelength from the optical transmission and reflection spectra. Subsequently, several models can be tested on the obtained data, in order to check their validity with respect to the materials under study. Moreover, these models offer the possibility to estimate essential physical parameters. Such models are tested within this article for the refraction index and for the real part of the complex dielectric constant, for bismuth trioxide thin films deposited on glass substrates maintained at three different temperatures. Also, the model proposed by Tauc is applied for the absorption spectrum of the same films, in order to determine the type of electronic transition and to estimate the optical energy bandgap. It will be noticed that the optical parameters vary rather significantly with changing substrate temperature, while the structure of the films, as studied by means of X‐ray diffractometry is almost insensitive to this change of deposition parameter. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)