On the Caginalp phase-field system based on type III with two temperatures and nonlinear coupling.

This paper is devoted to the study of a generalization of the Caginalp phase-field system based on the theory of type III thermomechanics with two temperatures for the heat conduction with a nonlinear coupling term. We start our analysis by establishing existence of the solutions. Then, we discuss dissipativity and uniqueness of the solutions. We finish our analysis by studying the spatial behavior of the solutions in a semi-infinite cylinder, assuming the existence of such solutions.     

Well-posedness and long time behavior of a hyperbolic Caginalp phase-field system

The aim of this paper is to prove the continuity of exponential attractors for a hyperbolic perturbed Caginalp system to an exponential attractor for the limit parabolichyperbolic Caginalp system. The symmetric distance between the perturbed and unperturbed exponential attractors in terms of the perturbation parameter is obtained.     

Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials

Our aim in this article is to prove the global (in time) existence of solutions to a Caginalp phase-field system with dynamic boundary conditions and a singular potential. The main difficulty is to prove that the solutions are strictly separated from the singular values of the potential. This is achieved by studying an auxiliary elliptic problem.     

Energy decay result in a Timoshenko‐type system of thermoelasticity of type III with weak damping

We investigate in this paper a thermoelastic system where the oscillations are defined by the Timoshenko model and the heat conduction is given by Green and Naghdi theories. We introduce 2 new stability numbers κ₁ , κ₂, and we prove a general decay result, from which the exponential and polynomial decays are only special cases.     

Asymptotic behaviour of a two‐dimensional differential system with non‐constant delay

The asymptotic behaviour and stability properties are studied for a real two‐dimensional system x^′(t) = A(t)x (t) + B(t)x (θ (t)) + h (t, x (t), x (θ (t))), with a nonconstant delay t ‐ θ (t) ≥ 0. It is supposed that A,B and h are matrix functions and a vector function, respectively. The method of investigation is based on the transformation of the considered real system to one equation with complex‐valued coefficients. Stability and asymptotic properties of this equation are studied by means of a suitable Lyapunov‐Krasovskii functional. The results generalize the great part of the results of J. Kalas and L. Baráková [J. Math. Anal. Appl. 269 , No. 1, 278–300 (2002)] for two‐dimensional systems with a constant delay (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Caginalp system with dynamic boundary conditions and singular potentials

This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. We first show that, for initial data in H ², the solutions are strictly separated from the singularities of the potential. This turns out to be our main argument in the proof of the existence and uniqueness of solutions. We then prove the existence of global attractors. In the last part of the article, we adapt well-known results concerning the Lojasiewicz inequality in order to prove the convergence of solutions to steady states.      

Energy decay rates for a Timoshenko-type system of thermoelasticity of type III with constant delay

In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with delay, where the heat conduction is given by Green and Naghdi theory. We establish the stability of the system for the case of equal and nonequal speeds of wave propagation.     

Some Qualitative Properties for the Total Variation Flow

We prove the existence of a finite extinction time for the solutions of the Dirichlet problem for the total variation flow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in finite time. The asymptotic profile of the solutions of the Dirichlet problem is also studied. It is shown that the profiles are nonzero solutions of an eigenvalue-type problem that seems to be unexplored in the previous literature. The propagation of the support is analyzed in the radial case showing a behaviour entirely different to the case of the problem associated with the p-Laplacian operator. Finally, the study of the radially symmetric case allows us to point out other qualitative properties that are peculiar of this special class of quasilinear equations.     

A stability result for the vibrations given by the standard linear model with thermoelasticity of type III

The aim of this article is to establish the well-posedness as well as an exponential stability result for the standard linear solid vibrating systems of thermoelasticity type III. Numerical experiments using finite differences are given to confirm our analytical result.     

Solutions of the generalized Lennard-Jones system

In this paper, we study solution structures of the following generalized Lennard-Jones system in R~n,x=(-α/|x|~(α+2)+β/|x|~(β+2))x,with 0 α β. Considering periodic solutions with zero angular momentum, we prove that the corresponding problem degenerates to 1-dimensional and possesses infinitely many periodic solutions which must be oscillating line solutions or constant solutions. Considering solutions with non-zero angular momentum, we compute Morse indices of the circular solutions first, and then apply the mountain pass theorem to show the existence of non-circular solutions with non-zero topological degrees. We further prove that besides circular solutions the system possesses in fact countably many periodic solutions with arbitrarily large topological degree, infinitely many quasi-periodic solutions, and infinitely many asymptotic solutions.     

Exponential stabilization of thermoelastic system of type II with non-uniform bounded disturbance

The boundary stabilization problem of a thermoelastic system of type II with a tip-mass is considered with the assumption that there is an external non-uniform bounded disturbance at the control end. In order to estimate this kind of disturbance, a time-varying high-gain estimator is designed, where the idea of active disturbance rejection control is adopted. Based on the estimate of disturbance, we propose a boundary feedback controller so as to exponentially stabilize this system. Finally, some numerical simulations on the dynamical behavior of the closed-loop system are given.     

On the asymptotics of some difference equations

We prove two inclusion theorems regarding a difference equation and apply them in getting the asymptotics of positive solutions of some concrete difference equations.     

Global well-posedness for the viscous shallow water system with Korteweg type

In this paper, we study the Cauchy problem for a viscous shallow water system with Korteweg type in Sobolev spaces. We first establish the local well-posedness of the solution by using the Friedrich method and compactness arguments. Then, we prove the global existence of the solution to the system for the small initial data.     

Minimal wave speed of a competition integrodifference system

In this paper, we study the minimal wave speed of a competitive system. By constructing upper and lower solutions, we confirm the existence of travelling wave solution at the critical wave speed. This completes earlier results found in the literature. Our conclusion implies that the asymptotic decay behaviour of solutions at the critical wave speed is different from that of solutions at larger wave speeds.     

Well-posedness and asymptotic stability to a laminated beam in thermoelasticity of type III

This paper is concerned with the well-posedness and asymptotic behaviour of solutions to a laminated beam in thermoelasticity of type III. We first obtain the well-posedness of the system by using semigroup method. We then investigate the asymptotic behaviour of the system through the perturbed energy method. We prove that the energy of system decays exponentially in the case of equal wave speeds and decays polynomially in the case of nonequal wave speeds. Under the case of nonequal wave speeds, we also investigate the lack of exponential stability of the system.     

On the Asymptotic Behaviour of Some Nonlocal Problems

The goal of this paper is to study diffusion problems associated with nonlinear diffusions of nonlocal type. We give existence and uniqueness results for these kind of problems and investigate the asymptotic behaviour.