Local spectra and individual stability of uniformly bounded -semigroups

We study the asymptotic behaviour of individual orbits of a uniformly bounded -semigroup with generator in terms of the singularities of the local resolvent on the imaginary axis. Among other things we prove individual versions of the Arendt-Batty-Lyubich-Vu theorem and the Katznelson-Tzafriri theorem.

     

Applications of Tauberian theorems in some problems in mathematical physics

We give some multidimensional Tauberian theorems for generalized functions and show examples of their application in mathematical physics. In particular, we consider the problems of stabilizing the solutions of the Cauchy problem for the heat kernel equation, multicomponent gas diffusion, and the asymptotic Cauchy problem for a free Schrödinger equation in the norms of different Banach spaces among others.     

Spectral stability criterion and the Cauchy problem for the Hill equation at parametric resonance

An analytical solution to the Cauchy problem for the Hill equation is constructed by the second-order averaging method for three instability domains, stability domains near the boundaries with the instability domains, and on the boundaries themselves. An unstable exponentially decaying solution is found in the instability domains. A simple (convenient for applications) stability criterion for the trivial solution is formulated in the form of an inequality expressed in terms of the constant component, the amplitudes, and the frequencies of harmonics in the spectrum of the periodic coefficient of the Hill equation.     

Wavelets and regularization of the Cauchy problem for the Laplace equation

In this paper, a Cauchy problem for two-dimensional Laplace equation in the strip 0<x?1 is considered again. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data, a small perturbation in the data can cause a dramatically large error in the solution for 0<x?1. The stability of the solution is restored by using a wavelet regularization method. Moreover, some sharp stable estimates between the exact solution and its approximation in Hr(R)-norm is also provided.     

The a posteriori Fourier method for solving the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data

In the present paper, the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data in an infinite “strip” domain is considered. This problem is severely ill-posed, i.e., the solution does not depend continuously on the data. A conditional stability result is given. A new a posteriori Fourier method for solving this problem is proposed. The corresponding error estimate between the exact solution and its regularization approximate solution is also proved. Numerical examples show the effectiveness of the method and the comparison of numerical effect between the a posteriori and the a priori Fourier method are also taken into account.     

A variational-type method of fundamental solutions for a Cauchy problem of Laplace’s equation

In this paper, we propose a new regularization method based on a finite-dimensional subspace generated from fundamental solutions for solving a Cauchy problem of Laplace’s equation in a simply-connected bounded domain. Based on a global conditional stability for the Cauchy problem of Laplace’s equation, the convergence analysis is given under a suitable choice for a regularization parameter and an a-priori bound assumption to the solution. Numerical experiments are provided to support the analysis and to show the effectiveness of the proposed method from both accuracy and stability.     

Abelian and Tauberian theorems for the Laplace transform of functions in several variables

Using two kinds of multivariate regular variation we prove several Abel-Tauber theorems for the Laplace transform of functions in several variables. We generalize some power series results of Alpar and apply our results in multivariate renewal theory.     

Trefftz energy method for solving the Cauchy problem of the Laplace equation

The inverse Cauchy problem of Laplace equation is hard to solve numerically, since it is highly ill-posed in the Hadamard sense. With this in mind, we propose a natural regularization technique to overcome the difficulty. In the linear space of the Trefftz bases for solving the Laplace equation, we introduce a novel concept to construct the Trefftz energy bases used in the numerical solution for the inverse Cauchy problem of the Laplace equation in arbitrary star plane domain. The Trefftz energy bases not only satisfy the Laplace equation but also preserve the energy, whose performance is better than the original Trefftz bases. We test the new method by two numerical examples.     

Harmonic Measures and Numerical Computation of Cauchy Problems for Laplace Equations*

It is well known that the Cauchy problem for Laplace equations is an ill-posed problem in Hadamard’s sense.Small deviations in Cauchy data may lead to large errors in the solutions.It is observed that if a bound is imposed on the solution,there exists a conditional stability estimate.This gives a reasonable way to construct stable algorithms.However,it is impossible to have good results at all points in the domain.Although numerical methods for Cauchy problems for Laplace equations have been wid...     

Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations

We study the Cauchy problem in the layer Π _T =ℝ ⁿ ×[0,T] for the equationu _t =cGΔu _t +Δϕ(u), wherec is a positive constant and the functionϕ(p) belongs toC ¹(ℝ₊) and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for the class of nonnegative functionsu(x,t) ∈C _x,t ^2,1 (Π _T ) with the properties: , . Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 356–362, September, 1996. This research was partially supported by the International Science Foundation under grant No. MX6000.     

On acyclicity of the set of solutions to the Cauchy problem for differential equations

The acyclicity of sets of solutions to the Cauchy problem is considered from the viewpoint of the axiomatic theory of solutions spaces of ordinary differential equations. We prove that the class of acyclic solution spaces is closed with respect to passing to the limit space. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 836–842, December, 1997 Translated by M. A. Shishkova     

A Trefftz/MFS mixed-type method to solve the Cauchy problem of the Laplace equation

By using the multiple-scale Trefftz method (MSTM) to solve the Cauchy problem of the Laplace equation in an arbitrary bounded domain, we may lose the accuracy several orders when the noise being imposed on the specified Cauchy data is quite large. In addition to the linear equations obtained from the MSTM, the fundamental solutions play as the test functions being inserted into a derived boundary integral equation. Therefore, after merely supplementing a few linear equations in the mixed-type method (MTM), which is a well organized combination of the Trefftz method and the method of fundamental solutions (MFS), we can improve the ill-conditioned behavior of the linear equations system and hence increase the accuracy of the solution for the Cauchy problem significantly, as explored by two numerical examples.     

Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations

The uniqueness and existence of BV-solutions for Cauchy problem of the form are proved.     

Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation

The paper studies the existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation arising in the model in variational form for the neo–Hookean elastomer rod where k₁, k₂>0 are real numbers, g(s) is a given nonlinear function. When g(s)=sⁿ (where n?2 is an integer), by using the Fourier transform method we prove that for any T>0, the Cauchy problem admits a unique global smooth solution u∈C^∞((0, T]; H^∞( R ))∩C([0, T]; H³( R ))∩C¹([0, T]; H^?1( R )) as long as initial data u₀∈W^{4, 1}( R )∩H³( R ), u₁∈L¹( R )∩H^?1( R ). Moreover, when (u₀, u₁)∈H²( R ) × L²( R ), g∈C²( R ) satisfy certain conditions, the Cauchy problem has no global solution in space C([0, T]; H²( R ))∩C¹([0, T]; L²( R ))∩H¹(0, T; H²( R )). Copyright © 2009 John Wiley & Sons, Ltd.     

The global weak solutions to the Cauchy problem of the generalized Novikov equation

In this paper, we study the Cauchy problem of the generalized Novikov equation. We first show that under suitable condition, the strong solution exists globally via some a priori estimates. Then, we prove the existence and uniqueness of global weak solutions by the approximation method. We also obtain the exact peaked solutions.     

Modulus of continuity of the coefficients and (non)quasianalytic solutions in the strictly hyperbolic Cauchy problem

In the strictly hyperbolic Cauchy problem, we investigate the relation between the modulus of continuity in the time variable of the coefficients and the well-posedness in Beurling-Roumieu classes of ultradifferentiable functions and functionals. We find well-posedness in nonquasianalytic classes assuming that the coefficients have modulus of continuity tω(1/t) such that . This condition is sharp because, in the case , we provide examples of Cauchy problems which are well-posed only in quasianalytic classes.     

Existence and uniqueness of holomorphic solutions for fractional Cauchy problem

By employing majorant functions, the existence and uniqueness of holomorphic solutions to nonlinear fractional partial differential equations (the Cauchy problems) are introduced. Furthermore, the analytic continuation of solutions is studied.     

Equality of two spectra arising in harmonic analysis and semigroup theory

We show that a new notion of a spectrum of a function ( is a Banach space), defined by B. Basit and the first author, coincides with the Arveson spectrum of some shift group, provided is uniformly continuous. We apply this result to prove a new version of a tauberian theorem.     

Fréchet differentiability of the solutions of a semilinear abstract Cauchy problem

Sufficient conditions are found under which the solutions z(t;q) of a semilinear abstract Cauchy problem of the form are Fréchet differentiable with respect to the parameter q. An explicit form is provided for the sensitivity equation satisfied by the Fréchet derivative Dqz(t;q).