Finite dimensionality and regularity of attractors for a 2-D semilinear wave equation with nonlinear dissipation

We consider a semilinear wave equation, defined on a two-dimensional bounded domain Ω, with a nonlinear dissipation. Our main result is that the flow generated by the model is attracted by a finite dimensional global attractor. In addition, this attractor has additional regularity properties that depend on regularity properties of nonlinear functions in the equation. To our knowledge this is a first result of this type in the context of higher dimensional wave equations.     

Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain

In this paper, we study the asymptotic behavior of solutions for the plate equation with a localized damping and a critical exponent. We prove the existence, regularity and finite dimensionality of a global attractor in .     

Quasi-rational function solutions of an elliptic equation and its application to solve some nonlinear evolution equations

Using the differential transformation method and the homogeneous balance method, some new solutions of an auxiliary elliptic equation are obtained. These solutions possess the forms of rational functions in terms of trigonometric functions, hyperbolic functions, exponential functions, power functions, elliptic functions and their operation and composite functions and so on, which are so-called quasi-rational function solutions. Based on these new quasi-rational functions solutions, a direct method is proposed to construct the exact solutions of some nonlinear evolution equations with the aid of symbolic computation. The coupled KdV-mKdV equation and Broer-Kaup equations are chosen to illustrate the effectiveness and convenience of the suggested method for obtaining quasi-rational function solutions of nonlinear evolution equations.     

Summatory and integral equations in periodic diffraction problems for elastic waves at defects in stratified media

We consider the diffraction problem for an elastic wave on a periodic set of defects located at the interface of stratified media. We reduce the mentioned problem to a pair summatory functional equation with respect to coefficients of the expansion of the desired wave by quasiperiodic waves (the Floquet waves). Using the method of integral identities, we reduce the pair equation to a regular infinite system of linear equations. One can solve this system by the truncation method. We prove that the integral identity is the necessary and sufficient condition for the solvability of the auxiliary overspecified problem for a system of equations in a half-plane in the elasticity theory. We obtain integral equations of the second kind which are equivalent to the initial diffraction problem.     

Asymptotic behavior of nonlinear sound waves in inviscid media with thermal and molecular relaxation

Nonlinear sound propagation through media with thermal and molecular relaxation can be modeled by third-order in time wave-like equations with memory. We investigate the asymptotic behavior of a Cauchy problem for such a model, the nonlocal Jordan–Moore–Gibson–Thompson equation, in the so-called critical case, which corresponds to propagation through inviscid fluids or gases. The memory has an exponentially fading character and type I, meaning that involves only the acoustic velocity potential. A major challenge in studying global behavior is that the linearized equation’s decay estimates are of regularity-loss type. As a result, the classical energy methods fail to work for the nonlinear problem. To overcome this difficulty, we construct appropriate time-weighted norms, where weights can have negative exponents. These problem-tailored norms create artificial damping terms that help control the nonlinearity and the loss of derivatives, and ultimately allow us to discover the model’s asymptotic behavior.     

Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems

We consider elliptic operators A on a bounded domain, that are compact perturbations of a selfadjoint operator. We first recall some spectral properties of such operators: localization of the spectrum and resolvent estimates. We then derive a spectral inequality that measures the norm of finite sums of root vectors of A through an observation, with an exponential cost. Following the strategy of Lebeau and Robbiano (1995) [25], we deduce the construction of a control for the non-selfadjoint parabolic problem t_∂u+Au=Bg. In particular, the L² norm of the control that achieves the extinction of the lower modes of A is estimated. Examples and applications are provided for systems of weakly coupled parabolic equations and for the measurement of the level sets of finite sums of root functions of A.     

Combined higher order finite volume and finite element scheme for double porosity and non-linear adsorption of transport problem in porous media

In this work, a dual porosity model of reactive solute transport in porous media is presented. This model consists of a nonlinear-degenerate advection-diffusion equation including equilibrium adsorption to the reaction combined with a first-order equation for the non-equilibrium adsorption interaction processes. The numerical scheme for solving this model involves a combined high order finite volume and finite element scheme for approximation of the advection-diffusion part and relaxation-regularized algorithm for nonlinearity-degeneracy. The combined finite volume-finite element scheme is based on a new formulation developed by Eymard et al. (2010) [10]. This formulation treats the advection and diffusion separately. The advection is approximated by a second-order local maximum principle preserving cell-vertex finite volume scheme that has been recently proposed whereas the diffusion is approximated by a finite element method. The result is a conservative, accurate and very flexible algorithm which allows the use of different mesh types such as unstructured meshes and is able to solve difficult problems. Robustness and accuracy of the method have been evaluated, particularly error analysis and the rate of convergence, by comparing the analytical and numerical solutions for first and second order upwind approaches. We also illustrate the performance of the discretization scheme through a variety of practical numerical examples. The discrete maximum principle has been proved.     

Global well-posedness of partially periodic KP-I equation in the energy space and application

In this article, we address the Cauchy problem for the KP-I equation

${?}_{t}u+{?}_x^{3}u?{?}_x^{?1}{?}_y^{2}u+u{?}_{x}u=0$

for functions periodic in y. We prove global well-posedness of this problem for any data in the energy space $\mathbf{E}=\{u\in L^2(\mathbb{R}\times \mathbb{T}),{?}_{x}u\in L^2(\mathbb{R}\times \mathbb{T}),{?}_x^{?1}{?}_{y}u\in L^2(\mathbb{R}\times \mathbb{T})\}$. We then prove that the KdV line soliton, seen as a special solution of KP-I equation, is orbitally stable under this flow, as long as its speed is small enough.     

On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation

We prove the global existence and uniqueness of admissible weak solutions to an asymptotic equation of a nonlinear hyperbolic variational wave equation with nonnegative L ²(ℝ) initial data. The work of Ping Zhang is supported by the Chinese postdoctor’s foundation, and that of Yuxi Zheng is supported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fellows award.     

A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order

Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schrödinger equation on the real line are studied in Sobolev spaces Hs, for s negative but close to 0. For smooth solutions there is an a priori upper bound for the Hs norm of the solution, in terms of the Hs norm of the datum, for arbitrarily large data, for sufficiently short time. Weak solutions are constructed for arbitrary initial data in Hs.     

Explicit secular equations of Rayleigh waves in elastic media under the influence of gravity and initial stress

The problem of Rayleigh waves in an orthotropic elastic medium under the influence of gravity and initial stress was investigated by Abd-Alla [A. M. Abd-Alla, Propagation of Rayleigh waves in an elastic half-space of orthotropic material, Appl. Math. Comput. 99 (1999) 61-69], and the secular equation of the wave in the implicit form was derived. However, due to the uncorrect representation of the solution, the secular equation is not right. The main aim of the present paper is to reconsider this problem. We find the secular equation of the wave in explicit form. By considering some special cases, we obtain the exact explicit secular equations of Rayleigh waves under the effect of gravity of some previous studies, in which only implicit secular equations were derived.     

Solution of boundary and eigenvalue problems for second‐order elliptic operators in the plane using pseudoanalytic formal powers

We propose a method for solving boundary value and eigenvalue problems for the elliptic operator D = div p grad + qin the plane using pseudoanalytic function theory and in particular pseudoanalytic formal powers. Under certain conditions on the coefficients p and q with the aid of pseudoanalytic function theory a complete system of null solutions of the operator can be constructed following a simple algorithm consisting in recursive integration. This system of solutions is used for solving boundary value and spectral problems for the operator D in bounded simply connected domains. We study theoretical and numerical aspects of the method. Copyright © 2010 John Wiley & Sons, Ltd.     

On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary

We study the asymptotic behavior of the solution of a 3D hyperbolic system arising in the Green-Naghdi models of thermoelasticity of type II and III with a dissipative boundary condition for the displacement and prove that the energy exponentially decays in time.     

On the selection of auxiliary functions, operators, and convergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach

The Homotopy Analysis Method of Liao [Liao SJ. Beyond perturbation: introduction to the Homotopy Analysis Method. Boca Raton: Chapman & Hall/CRC Press; 2003] has proven useful in obtaining analytical solutions to various nonlinear differential equations. In this method, one has great freedom to select auxiliary functions, operators, and parameters in order to ensure the convergence of the approximate solutions and to increase both the rate and region of convergence. We discuss in this paper the selection of the initial approximation, auxiliary linear operator, auxiliary function, and convergence control parameter in the application of the Homotopy Analysis Method, in a fairly general setting. Further, we discuss various convergence requirements on solutions.     

Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves

The time asymptotic behavior of a solution to the initial Cauchy problem for a quasilinear parabolic equation is investigated. Such equations arise, for example, in traffic flow modeling. The main result of this paper is the proof of the previously formulated conjecture that, if a monotone initial function has limits at plus and minus infinity, then the solution to the Cauchy problem converges in form to a system of traveling and rarefaction waves; furthermore, the phase shifts of the traveling waves may depend on time. It is pointed out that the monotonicity condition can be replaced with the boundedness condition.     

The L-L estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media

We first obtain the L^p-L^q estimates of solutions to the Cauchy problem for one-dimensional damped wave equation

     

Reflection and transmission of plane acoustic waves in an infinite annular duct with a finite gap on the inner wall

The diffraction of acoustic waves by an infinitely long annular duct having a finite gap on the inner wall is investigated rigorously. The related boundary‐value problem is formulated into a modified Wiener–Hopf equation, which is then reduced to a pair of simultaneous Fredholm integral equations of the second kind. At the end of the analysis, numerical results illustrating the effects of the width of the coaxial cylindrical waveguide and the gap length on the diffraction phenomenon are presented. Copyright © 2010 John Wiley & Sons, Ltd.