Damped wave equation with a critical nonlinearity

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity

where 0,$"> and space dimensions . Assume that the initial data

where \frac{n}{2},$"> weighted Sobolev spaces are

Also we suppose that

0,\int u_{0}\left( x\right) dx>0, \end{displaymath}">

where

Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property

for all 0,$"> where

     

Estimates of

We extend a result of Ramaré and Rumely, 1996, about the Chebyshev function in arithmetic progressions. We find a map such that and 0)}$">, whereas is a constant. Now we are able to show that, for ,

and, for ,

\frac{x}{2\ln x}.\end{displaymath}">

     

Multivariable Bohr inequalities

Operator-valued multivariable Bohr type inequalities are obtained for:

(i)

a class of noncommutative holomorphic functions on the open unit ball of , generalizing the analytic functions on the open unit disc;

(ii)

the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra ;

(iii)

a class of noncommutative selfadjoint harmonic functions on the open unit ball of , generalizing the real-valued harmonic functions on the open unit disc;

(iv)

the Cuntz-Toeplitz algebra , the reduced (resp. full) group -algebra (resp.  ) of the free group with generators;

(v)

a class of analytic functions on the open unit ball of .

The classical Bohr inequality is shown to be a consequence of Fejér's inequality for the coefficients of positive trigonometric polynomials and Haager- up-de la Harpe inequality for nilpotent operators. Moreover, we provide an inequality which, for analytic polynomials on the open unit disc, is sharper than Bohr's inequality.

     

A global approach to fully nonlinear parabolic problems

We consider the general initial-boundary value problem

(1)        
(2)        
(3)        
where is a bounded open set in with sufficiently smooth boundary.  The problem (1)-(3) is first reduced to the analogous problem in the space with zero initial condition and

The resulting problem is then reduced to the problem where the operator satisfies Condition  This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces.  The local and global solvability of the operator equation are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.

     

Unbounded solutions and periodic solutions for second order differential equations with asymmetric nonlinearity

In this paper we will prove the coexistence of unbounded solutions and periodic solutions for the asymmetric oscillator

where and are positive constants satisfying the nonresonant condition

and is periodic in the first variable and bounded.

     

Bochner-Riesz means with respect to a rough distance function

The generalized Bochner-Riesz operator may be defined as

where is an appropriate distance function and is the inverse Fourier transform. The behavior of on is described for , a rough distance function. We conjecture that this operator is bounded on when and , and unbounded when . This conjecture is verified for large ranges of .

     

Inequalities for eigenvalues of a clamped plate problem

Let be a connected bounded domain in an -dimensional Euclidean space . Assume that

are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator:

Then, we give an upper bound of the -th eigenvalue in terms of the first eigenvalues, which is independent of the domain , that is, we prove the following:

Further, a more explicit inequality of eigenvalues is also obtained.

     

Hyperbolic conservation laws with stiff reaction terms of monostable type

In this paper the zero reaction limit of the hyperbolic conservation law with stiff source term of monostable type

is studied. Solutions of Cauchy problems of the above equation with initial value are proved to converge, as , to piecewise constant functions. The constants are separated by either shocks determined by the Rankine-Hugoniot jump condition, or a non-shock jump discontinuity that moves with speed . The analytic tool used is the method of generalized characteristics. Sufficient conditions for the existence and non-existence of traveling waves of the above system with viscosity regularization are given. The reason for the failure to capture the correct shock speed by first order shock capturing schemes when underresolving 0$"> is found to originate from the behavior of traveling waves of the above system with viscosity regularization.

     

The linear heat equation with highly oscillating potential

In this paper we consider the following initial value problem:

where and . Nonexistence of positive solutions is analyzed.

     

``Best possible' upper and lower bounds for the zeros of the Bessel function

Let denote the -th positive zero of the Bessel function . In this paper, we prove that for and , 2, 3, ,

These bounds coincide with the first few terms of the well-known asymptotic expansion

as , being fixed, where is the -th negative zero of the Airy function , and so are ``best possible'.

     

Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains

We consider the operator

acting on functions in . We prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on and . In contrast to previous work, the need only be nonnegative on the boundary rather than strictly positive, at the expense of the and being Hölder continuous. Applications to super-Markov chains are given. The proof follows Stroock and Varadhan's perturbation argument, but the underlying function space is now a weighted Hölder space and each component of the constant coefficient process being perturbed is the square of a Bessel process.

     

Oscillation criteria for delay equations

This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form

(1)

where is non-decreasing, for and . Let the numbers and be defined by

It is proved here that when and all solutions of Eq. (1) oscillate in several cases in which the condition

2k+\frac{2}{{\lambda}_{1}}-1 \end{displaymath}">

holds, where is the smaller root of the equation .

     

Gap estimates of the spectrum of Hill's equation and action variables for

Consider the Schrödinger equation for a potential of period 1 in the weighted Sobolev space

where denote the Fourier coefficients of when considered as a function of period 1,

and where is the circle of length 1. Denote by the periodic eigenvalues of when considered on the interval with multiplicities and ordered so that We prove the following result. Theorem. For any bounded set there exist and so that for and , the eigenvalues are isolated pairs, satisfying (with

(i)

,

(ii)

.

     

A nonlinear Fokker-Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma

This work deals with the problem consisting in the equation

together with no-flux conditions at and , i.e.

Such a problem arises as a kinetic approximation to describe the evolution of the radiation distribution in a homogeneous plasma when radiation interacts with matter via Compton scattering. We shall prove that there exist solutions of , which develop singularities near in a finite time, regardless of how small the initial number of photons is. The nature of such singularities is then analyzed in detail. In particular, we show that the flux condition is lost at when the singularity unfolds. The corresponding blow-up pattern is shown to be asymptotically of a shock wave type. In rescaled variables, it consists in an imploding travelling wave solution of the Burgers equation near , that matches a suitable diffusive profile away from the shock. Finally, we also show that, on replacing near as determined by the manner of blow-up, such solutions can be continued for all times after the onset of the singularity.

     

A multiplicity theorem for the Neumann problem

Here is a particular case of the main result of this paper: Let be a bounded domain, with a boundary of class , and let be two continuous functions, , with 0$">, , with n$">. If

and if the set of all global minima of the function has at least connected components, then, for each 0$"> small enough, the Neumann problem

admits at least strong solutions in .

     

One-dimensional asymptotic classes of finite structures

A collection of finite -structures is a 1-dimensional asymptotic class if for every and every formula , where :

(i)

There is a positive constant and a finite set such that for every and , either , or for some ,

(ii)

For every , there is an -formula , such that is precisely the set of with

One-dimensional asymptotic classes are introduced and studied here. These classes come equipped with a notion of dimension that is intended to provide for the study of classes of finite structures a concept that is central in the development of model theory for infinite structures. Connections with the model theory of infinite structures are also drawn.

     

The singular limit of a vector-valued reaction-diffusion process

We study the asymptotic behaviour of the solution to the vector-valued reaction-diffusion equation

where . We assume that the the potential depends only on the modulus of and vanishes along two concentric circles. We present a priori estimates for the solution , and, in the spatially radially symmetric case, we show rigorously that in the singular limit as , two phases are created. The interface separating the bulk phases evolves by its mean curvature, while evolves according to a harmonic map flow on the respective circles, coupled across the interfaces by a jump condition in the gradient.

     

Combinatorial congruences modulo prime powers

Let be any prime, and let and be nonnegative integers. Let and . We establish the congruence

(motivated by a conjecture arising from algebraic topology) and obtain the following vast generalization of Lucas' theorem: If is greater than one, and are nonnegative integers with , then

We also present an application of the first congruence to Bernoulli polynomials and apply the second congruence to show that a -adic order bound given by the authors in a previous paper can be attained when .

     

On Littlewood's boundedness problem for sublinear Duffing equations

In this paper, we are concerned with the boundedness of all the solutions and the existence of quasi-periodic solutions for second order differential equations

where the 1-periodic function is a smooth function and satisfies sublinearity:

     

Optimal constants in the exceptional case of Sobolev inequalities on Riemannian manifolds

Let be a Riemannian compact -manifold. We know that for any , there exists such that for any , , being the smallest constant possible such that the inequality remains true for any . We call the ``first best constant'. We prove in this paper that it is possible to choose and keep a finite constant. In other words we prove the existence of a ``second best constant' in the exceptional case of Sobolev inequalities on compact Riemannian manifolds.