Spatial estimates for an equation with a delay term

In this note, we investigate the spatial behaviour of the solutions for a theory for the heat conduction with a delay term. We obtain an alternative of the Phragmen–Lindelof type. That is the solutions either decay in a exponential way or blow-up at infinity in a exponential way. We also describe how to obtain an upper bound for the amplitude term. It is worth noting that this is the first contribution on spatial behaviour for partial differential equations involving a delay term. We use the energy arguments to obtain our main results. The main point of the contribution is the use of a suitable weighted energy function.     

A note on a non-standard problem for an equation with a delay term

In this note we investigate the continuous dependence of the solutions for a theory of heat conduction with a delay term. We use energy arguments to obtain the continuous dependence results and spectral arguments to prove the non-uniqueness result. The extension to the thermoelastic problem is also pointed out.     

The heat flow with a critical exponential nonlinearity

We analyze the possible concentration behavior of heat flows related to the Moser–Trudinger energy and derive quantization results completely analogous to the quantization results for solutions of the corresponding elliptic equation. As an application of our results we obtain the existence of critical points of the Moser–Trudinger energy in a supercritical regime.     

Variational approximations in energy space for the regulator problem with time delay

We consider a variational procedure for approximating the solution of the state regulator problem with time delay. Motivated by a dual formulation of the problem, we introduce a positive-definite functionalF over a certain energy space of Mikhlin and obtain approximating solutions by the Ritz-Trefftz idea of minimizing it over finite-dimensional subspaces. The resulting approximating solutions, in turn, furnish suboptimal solutions which converge to the optimal solution of the regulator problem with time delay. A priori error bounds in terms of splines are given. A posteriori error bounds are also obtained.     

Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations

We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg–deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained.     

Finite element Galerkin methods for multi-phase stefan problems

The modelling of heat conduction with phase transitions in terms of integral indentities in Galerkin form is actually based directly on physical concepts like conservation of heat, the conduction law etc. In this way we obtain a formulation which apparently covers general physical situations, for instance with melting and freezing of layered materials. Choosing finite elements in a fixed grid we can still track the moving boundaries by an accurate treatment of the nonlinear enthalpy terms. This makes it possible to use standard finite element packages to build an efficient programme for Stefan-like problems.     

Properties of stationary states of delay equations with large delay and applications to laser dynamics

We consider properties of periodic solutions of the differential‐delay system, which models a laser with optical feedback. In particular, we describe a set of multipliers for these solutions in the limit of large delay. As a preliminary result, we obtain conditions for stability of an equilibrium of a generic differential‐delay system with fixed large delay τ. We also show a connection between characteristic roots of the equilibrium and multipliers of the mapping obtained via the formal limit τ→∞. Copyright © 2004 John Wiley & Sons, Ltd.     

Estimates of solutions to a class of systems of nonlinear delay differential equations

Under study are the systems of nonlinear delay differential equations with periodic coefficients of the linear terms. Some sufficient conditions for the asymptotic stability of the zero solution are established. We obtain the estimates that characterize the decay rate of solutions at infinity and describe the attraction sets of the zero solution.     

On the Spatial and Temporal Behavior in Dynamics of Porous Elastic Mixtures

In this paper, we study the spatial and temporal behavior of dynamic processes in porous elastic mixtures. For the spatial behavior, we use the time-weighted surface power function method in order to obtain a more precise determination of the domain of influence and establish spatial-decay estimates of the Saint-Venant type with respect to time-independent decay rate for the inside of the domain of influence. For the asymptotic temporal behavior, we use the Cesáro means associated with the kinetic and strain energies and establish the asymptotic equipartition of the total energy. A uniqueness theorem is proved for finite and infinite bodies, and we note that it is free of any kind of a priori assumptions on the solutions at infinity.     

Quantitative uniqueness for elliptic equations with singular lower order terms

We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second-order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.     

Global Dispersive Solutions for the Gross–Pitaevskii Equation in Two and Three Dimensions

We study asymptotic behaviour at time infinity of solutions close to the non-zero constant equilibrium for the Gross–Pitaevskii equation in two and three spatial dimensions. We construct a class of global solutions with prescribed dispersive asymptotic behavior, which is given in terms of the linearized evolution. Submitted: May 24, 2006. Revised: December 21, 2006. Accepted: February 6, 2007.     

脉冲时滞微分方程解的整体存在唯一性、振动性与非振动性

本文讨论脉冲时滞微分方程X’(t)=f(t,x(t-T_1(t)),…,x(t-T_n(t))),x(t_k)-x(t_k~-)=I_k(x(t_k~- )).获得了方程(E) 解的一个整体存在唯一性定理.当(E)是线性方程时,给出了由时滞微分方程解的振动性或非振动性刻划出相应的脉冲时滞微分方程的同样性质的一般性脉冲条件.     

Spatial behaviour of solutions of the three-phase-lag heat equation

In this paper we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. The theory of three-phase-lag heat conduction leads to a hyperbolic partial differential equation with a fourth-order derivative with respect to time. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under a boundedness restriction on the initial data, an energy estimate is obtained. An upper bound for the amplitude term in this estimate in terms of the initial and boundary data is also established. For the case of zero initial conditions, a more explicit estimate is obtained which shows that solutions decay exponentially along certain spatial-time lines. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T₀ are assumed proportional to their initial values. These results are relevant in the context of the Saint-Venant Principle for heat conduction problems.     

Rough Isometry and <Emphasis Type="Italic">p</Emphasis>-Harmonic Boundaries of Complete Riemannian Manifolds

In this paper, we describe the behavior of bounded energy finite solutions for certain nonlinear elliptic operators on a complete Riemannian manifold in terms of its p-harmonic boundary. We also prove that if two complete Riemannian manifolds are roughly isometric to each other, then their p-harmonic boundaries are homeomorphic to each other. In the case, there is a one to one correspondence between the sets of bounded energy finite solutions on such manifolds. In particular, in the case of the Laplacian, it becomes a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral on the manifolds. This work was supported by grant No. R06-2002-012-01001-0(2002) from the Basic Research Program of the Korea Science & Engineering Foundation.     

On the spatial behavior in Type III thermoelastodynamics

This note is concerned with the linear (and linearized) Type III thermoelastodynamic theory proposed by Green and Naghdi. We investigate the spatial behavior of the solutions when we assume the positivity of the elasticity tensor, the thermal conductivity tensor, the mass density and the heat capacity. However, we do not assume (a priori) the positivity of the internal energy. We first obtain a Phragmén-Lindelöf alternative of exponential type for the solutions. Later, we prove that the decay can be controlled by the exponential of a second-degree polynomial. This is similar to other thermoelastic situations.     

基于MFS的PDE-约束优化法求解热传导逆问题

在本文中,我们给出了一种有效的无网格方法来求解逆热传导问题,含有Neumann边界条件情形.所得到的PDE-约束优化法是一种在空间与时间域上的全局近似方法,其中将控制方程的基本解作为基函数.由于初始测量数据包含有噪声误差,则所得线性方程组的系数矩阵通常是病态的,文中利用广义交叉验证(GCV)的Tikhonov正则化方法来获得更加稳定的数值解.通过数值结果表明,本文给出的方法是精确、有效、鲁棒的.     

Global nonexistence for logarithmic wave equations with nonlinear damping and distributed delay terms

This paper is concerned with global nonexistence of solutions for a logarithmic wave equation with nonlinear damping and distributed delay terms. Due to the simultaneous presence of nonlinear damping and logarithmic source terms, we have difficulty in use of the concavity method. Applying the energy estimates, we show the global nonexistence of solutions with not only non-positive initial energy but also positive initial energy.     

On uniqueness and analyticity in thermoviscoelastic solids with voids

In this paper we consider the most general system proposed&nbsp;to describe the thermoviscoelasticity with voids. We study two qualitative&nbsp;properties of the solutions of this theory. First, we obtain a&nbsp;uniqueness result when we do not&nbsp;assume any sign to the internal energy.&nbsp;Second we extend some previous results and prove the analyticity&nbsp;of the solutions. The impossibility of localization in time&nbsp;of the solutions&nbsp;is a consequence. Last result we present corresponds to the analyticity&nbsp;of solutions in case that the dissipation is not very strong, but with&nbsp;suitable coupling terms.     

Spatial behaviour of solutions of the dual‐phase‐lag heat equation

In this paper we study the spatial behaviour of solutions of some problems for the dual‐phase‐lag heat equation on a semi‐infinite cylinder. The theory of dual‐phase‐lag heat conduction leads to a hyperbolic partial differential equation with a third derivative with respect to time. First, we investigate the spatial evolution of solutions of an initial boundary‐value problem with zero boundary conditions on the lateral surface of the cylinder. Under a boundedness restriction on the initial data, an energy estimate is obtained. An upper bound for the amplitude term in this estimate in terms of the initial and boundary data is also established. For the case of zero initial conditions, a more explicit estimate is obtained which shows that solutions decay exponentially along certain spatial‐time lines. A class of non‐standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T are assumed proportional to their initial values. Copyright © 2004 John Wiley & Sons, Ltd.