Abstract and Applied Analysis

https://www.hindawi.com/journals/aaa/

List of Papers (Total 6,478)

Uniform stabilization of a coupled structural acoustic system by boundary dissipation

We consider a coupled PDE system arising in noise reduction problems. In a two dimensional chamber, the acoustic pressure (unwanted noise) is represented by a hyperbolic wave equation. The floor of the chamber is subject to the action of piezo-ceramic patches (smart materials). The goal is to reduce the acoustic pressure by means of the vibrations of the floor which is modelled...

Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation (d/dt)u(t)=Au(t)

On the discreteness of the spectra of the Dirichlet and Neumann p-biharmonic problems

We are interested in a nonlinear boundary value problem for (|u″|p−2u″)′​′=λ|u|p−2u in [0,1], p>1, with Dirichlet and Neumann boundary conditions. We prove that eigenvalues of the Dirichlet problem are positive, simple, and isolated, and form an increasing unbounded sequence. An eigenfunction, corresponding to the nth eigenvalue, has precisely n−1 zero points in (0,1...

Iterative solution of unstable variational inequalities on approximately given sets

The convergence and the stability of the iterative regularization method for solving variational inequalities with bounded nonsmooth properly monotone (i.e., degenerate) operators in Banach spaces are studied. All the items of the inequality (i.e., the operator A, the “right hand side” f and the set of constraints Ω) are to be perturbed. The connection between the parameters of...

Inertial manifolds and stabilization of nonlinear beam equations with Balakrishnan-Taylor damping

In this paper we study a hinged, extensible, and elastic nonlinear beam equation with structural damping and Balakrishnan-Taylor damping with the full exponent 2(n

An Ambrosetti-Prodi-type problem for an elliptic system of equations via monotone iteration method and Leray-Schauder degree theory

In this paper we employ the Monotone Iteration Method and the Leray-Schauder Degree Theory to study an ℝ2-parametrized system of elliptic equations. We obtain a curve dividing the plane into two regions. Depending on which region the parameter is, the system will or will not have solutions. This is an Ambrosetti-Prodi-type problem for a system of equations.

Carleson embeddings

In this paper we discuss several operator ideal properties for so called Carleson embeddings of tent spaces into specific L q(μ)-spaces, where μ is a Carleson measure on the complex unit disc. Characterizing absolutely q-summing, absolutely continuous and q-integral Carleson embeddings in terms of the underlying measure is our main topic. The presented results extend and...

The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics

We show here the uniform stabilization of a coupled system of hyperbolic and parabolic PDE's which describes a particular fluid/structure interaction system. This system has the wave equation, which is satisfied on the interior of a bounded domain Ω, coupled to a “parabolic–like” beam equation holding on ∂Ω, and wherein the coupling is accomplished through velocity terms on the...

Existence of multiple critical points for an asymptotically quadratic functional with applications

Morse theory for isolated critical points at infinity is used for the existence of multiple critical points for an asymptotically quadratic functional. Applications are also given for the existence of multiple nontrivial periodic solutions of asymptotically Hamiltonian systems.

Embedding functions and their role in interpolation theory

The embedding functions of an intermediate space A into a Banach couple (A0,A1) are defined as its embedding constants into the couples (1αA0,1βA1), ∀α,β>0. Using these functions, we study properties and interrelations of different intermediate spaces, give a new description of all real interpolation spaces, and generalize the concept of weak-type interpolation to any Banach...

Stability of coupled systems

The exponential and asymptotic stability are studied for certain coupled systems involving unbounded linear operators and linear infinitesimal semigroup generators. Examples demonstrating the theory are also given from the field of partial differential equations.

On quasilinear elliptic equations in ℝN

In this note we give a result for the operator p-Laplacian complementing a theorem by Brézis and Kamin concerning a necessary and sufficient condition for the equation −Δu=h(x)uq in ℝN, where 0<q<1, to have a bounded positive solution. While Brézis and Kamin use the method of sub and super solutions, we employ variational arguments for the existence of solutions.

Global attractors for two-phase stefan problems in one-dimensional space

In this paper we consider one-dimensional two-phase Stefan problems for a class of parabolic equations with nonlinear heat source terms and with nonlinear flux conditions on the fixed boundary. Here, both time-dependent and time-independent source terms and boundary conditions are treated. We investigate the large time behavior of solutions to our problems by using the theory for...

A convergence result for discreet steepest decent in weighted sobolev spaces

A convergence result is given for discrete descent based on Sobolev gradients arising from differential equations which may be expressed as quadratic forms. The argument is an extension of the result of David G. Luenberger on Euclidean descent and compliments the work of John W. Neuberger on Sobolev descent.

Evolution semigroups for nonautonomous Cauchy problems

In this paper, we characterize wellposedness of nonautonomous, linear Cauchy problems (NCP)  {u˙(t)=A(t)u(t)u(s)=x∈X on a Banach space X by the existence of certain evolution semigroups.

Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators

Let iAj(1≤j≤n) be generators of commuting bounded strongly continuous groups, A≡(A1,A2,…,An). We show that, when f has sufficiently many polynomially bounded derivatives, then there exist k,r>0 such that f(A) has a (1

Stable Approximations of a Minimal Surface Problem with Variational Inequalities

In this paper we develop a new approach for the stable approximation of a minimal surface problem associated with a relaxed Dirichlet problem in the space BV(Ω) of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional ? on BV(Ω) defined by ?(u)=?(u)

A new spectral theory for nonlinear operators and its applications

In this paper, by applying (p,k)-epi mapping theory, we introduce a new definition of spectrum for nonlinear operators which contains all eigenvalues, as in the linear case. Properties of this spectrum are given and comparison is made with the other definitions of spectra. We also give applications of the new theory.

Eigenvalues and ranges for perturbations of nonlinear accretive and monotone operators in Banach spaces

Various eigenvalue and range results are given for perturbations of m-accretive and maximal monotone operators. The eigenvalue results improve and extend some recent results by Guan and Kartsatos, while the range theorem gives an affirmative answer to a recent problem of Kartsatos.

N-Laplacian equations in ℝN with critical growth

We study the existence of nontrivial solutions to the following problem: {u∈W1,N(ℝN),u≥0  and−div(|∇u|N−2∇u)

Modelling of the Czochralski flow

The Czochralski method of the industrial production of a silicon single crystal consists of pulling up the single crystal from the silicon melt. The flow of the melt during the production is called the Czochralski flow. The mathematical description of the flow consists of a coupled system of six P.D.E. in cylindrical coordinates containing Navier-Stokes equations (with the stream...

Existence of positive radial solutions for a weakly coupled system via blow up

The existence of positive solutions to certain systems of ordinary differential equations is studied. Particular forms of these systems are satisfied by radial solutions of associated partial differential equations.

Uniform asymptotic normal structure, the uniform semi-Opial property and fixed points of asymptotically regular uniformly lipschitzian semigroups. Part I

In this paper we introduce the uniform asymptotic normal structure and the uniform semi-Opial properties of Banach spaces. This part is devoted to a study of the spaces with these properties. We also compare them with those spaces which have uniform normal structure and with spaces with WCS(X)>1.

Analyticity of thermo-elastic semigroups with coupled hinged/Neumann B.C.

We consider a thermo-elastic plate system where the elastic equation does not account for rotational forces. We select the case of hinged mechanical B.C. and Neumann thermal B.C., which are coupled on the boundary. We show that the corresponding s.c. contraction semigroup (on a natural energy space) is analytic and, hence, uniformly stable. Because of the boundary (high) coupling...

Multiple solutions for a problem with resonance involving the p-Laplacian

In this paper we will investigate the existence of multiple solutions for the problem (P)                                                         −Δpu