Convergence of the solution of a nonsymmetric matrix Riccati differential equation to its stable equilibrium solution

We consider the initial value problem for a nonsymmetric matrix Riccati differential equation, where the four coefficient matrices form an M-matrix. We show that for a wide range of initial values the Riccati differential equation has a global solution X(t) on [0,∞) and X(t) converges to the stable equilibrium solution as t goes to infinity.     

Nonlocal higher order evolution equations

In this article, we study the asymptotic behaviour of solutions to the nonlocal operator u _t (x, t) = (?1) ^n?1 (J * Id ? 1) ⁿ (u(x, t)), x ∈ ? ^N , which is the nonlocal analogous to the higher order local evolution equation v _t = (?1) ^n?1(Δ) ⁿ v. We prove that the solutions of the nonlocal problem converge to the solution of the higher order problem with the right-hand side given by powers of the Laplacian when the kernel J is rescaled in an appropriate way. Moreover, we prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity.     

Polynomial and exponential stability for one-dimensional problem in thermoelastic diffusion theory

We consider one-dimensional problem for the thermoelastic diffusion theory and we obtain polynomial decay estimates. Then we show that the solution decays exponentially to zero as time goes to infinity; that is, denoting by E(t) the first-order energy of the system, we show that positive constants C ₀ and c ₀ exist which satisfy E(t) ≤ C ₀ E(0)e ^{?c ₀ t} .     

Remark on asymptotic behaviors of solutions to a quasi-linear elliptic Dirichlet problem with large diffusion

We study asymptotic behaviors of nontrivial solutions to the Dirichlet problem of a quasi-linear elliptic equation and obtain a lower bound for growth of L^∞-norm of the solutions, which implies the L^∞-norm of the solutions goes to infinity as the diffusion coefficient goes to infinity.     

Convergence to equilibrium for a fully hyperbolic phase‐field model with Cattaneo heat flux law

We consider a fully hyperbolic phase‐field model in this paper. Our model consists of a damped hyperbolic equation of second order with respect to the phase function χ(t) , which is coupled with a hyperbolic system of first order with respect to the relative temperature θ(t) and the heat flux vector q (t). We prove the well‐posedness of this system subject to homogeneous Neumann boundary condition and no‐heat flux boundary condition. Then, we show that this dynamical system is a dissipative one. Finally, using the celebrated ?ojasiewicz–Simon inequality and by constructing an auxiliary functional, we prove that the solution of this problem converges to an equilibrium as time goes to infinity. We also obtain an estimate of the decay rate to equilibrium. Copyright © 2008 John Wiley & Sons, Ltd.     

Non-Diffusive Large Time Behavior for a Degenerate Viscous Hamilton–Jacobi Equation

The convergence to non-diffusive self-similar solutions is investigated for non-negative solutions to the Cauchy problem ? _t u = Δ _p u + |? u| ^q when the initial data converge to zero at infinity. Sufficient conditions on the exponents p > 2 and q > 1 are given that guarantee that the diffusion becomes negligible for large times and the L ^∞-norm of u(t) converges to a positive value as t → ∞.     

Lattice model for fast diffusion equation

We obtain a fast diffusion equation (FDE) as scaling limit of a sequence of zero-range process with symmetric unit rate. Fast diffusion effect comes from the fact that the diffusion coefficient goes to infinity as the density goes to zero. In order to capture this fast diffusion effect from a microscopic point of view we are led to consider a proper rescaling of a model with a typically high number of particles per site. Furthermore, we obtain some results on the convergence for the method of lines for FDE.     

Boundary behaviour for solutions of boundary blow-up problems in a borderline case

We investigate boundary blow-up solutions of the equation Δu=f(u) in a bounded domain Ω⊂RN under the condition that f(t) has a relatively slow growth as t goes to infinity. We show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x) in terms of the distance of x from ∂Ω.     

A limit theorem for perturbed operator semigroups with applications to random evolutions

Let U(t) and S(t) be strongly continuous contraction semigroups on a Banach space L with infinitesimal operators A and B, respectively. Suppose the closure of A + αB generates a semigroup T_α(t). The behavior of T_α(t) as α goes to infinity is examined. In particular, suppose S(t) converges strongly to P. If the closure of PA generates a semigroup T(t) on $\text{R}$(P), then T_α(t) goes to T(t) on $\text{R}$(P). If PA = 0 and if BVf = ?f for fε$\text{N}$(P), conditions are given that imply T_α(αt) converges on $\text{R}$(P) to a semigroup generated by the closure of PAVA.The results are used to obtain new and known limit theorems for random evolutions, which in turn give approximation theorems for diffusion processes.     

Asymptotic Behavior of Poisson Kernels on NA Groups

On a Lie group S = NA, that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, a probability measure μ is considered and treated as a distribution according to which transformations s ∈ S acting on N = S/A are sampled. Under natural conditions, formulated some over thirty years ago, there is a μ-invariant measure m on N. Properties of m have been intensively studied by a number of authors. The present article deals with the situation when μ(A) = ?(s _t  ∈ A), where ?⁺ ? t → s _t  ∈ S is the diffusion on S generated by a second order subelliptic, hypoelliptic, left-invariant operator on S. In this article the most general operators of this kind are considered. Precise asymptotic for m at infinity and for the Green function of the operator are given. To achieve this goal a pseudodifferential calculus for operators with coefficients of finite smoothness is formulated and applied.     

The Liar Game Over an Arbitrary Channel

We introduce and analyze a liar game in which t-ary questions are asked and the responder may lie at most k times. As an additional constraint, there is an arbitrary but prescribed list (the channel) of permissible types of lies. For any fixed t, k, and channel, we determine the exact asymptotics of the solution when the number of queries goes to infinity.     

Global existence and uniqueness of measure valued solutions to a Vlasov‐type equation with local alignment

We use a particle method to study a Vlasov‐type equation with local alignment, which was proposed by Sebastien Motsch and Eitan Tadmor [J. Statist. Phys., 141(2011), pp. 923‐947]. For N‐particle system, we study the unconditional flocking behavior for a weighted Motsch‐Tadmor model and a model with a “tail”. When N goes to infinity, global existence and stability (hence uniqueness) of measure valued solutions to the kinetic equation of this model are obtained. We also prove that measure valued solutions converge to a flock. The main tool we use in this paper is Monge‐Kantorovich‐Rubinstein distance.     

On sets of linear differential systems to which perturbed linear systems cannot be reduced

The paper [2] defines the noncoinciding irreducibility sets N ₂(a, σ) and N ₃(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) such that ‖A(t)‖ ≤ a < + ∞ for t ∈ [0,+∞) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B(t) satisfying [for the case of N ₂(a, σ)] the condition

On an autoregressive model with time-dependent coefficients

Summary As one of the non-stationary time series model, we consider a firstorder autoregressive model in which the autoregressive coefficient is assumed to be a function,f _t (θ), of timet. We establish several assumptions onf _t (θ), not on the terms in the Taylor expansion of log-likelihood function, and show that the estimators of unknown parameters involved inf _t (θ) have strong consistency and asymptotic normality under these assumptions when sample size tends to infinity.     

Uniform convergence of the horizontal line method for solutions and free boundaries in Stefan evolution inequalities

The change of variable for the temperature Θ in the one-phase Stefan problem leads to the evolution inequality, (u_t – Δu – f)(v – u) ? 0 for all regular v ? 0, where u ? 0 is required. This inequality is to hold over a space-time domain D = Ω × (0, T) with a Dirichlet boundary condition imposed on ? Ω × (0, T) and a zero initial condition. The free boundary phase interface is given in one space dimension by The fully implicit divided difference scheme leads to a sequence of elliptic variational inequalities for {u_m}. The sequence {u_m} may be interpolated linearly in t to obtain an approximation U_Δt of u. The following results are obtained in this paper: (i) a two-sided weak maximum principle for u_m – u_m-1 in N space dimensions, hence the free boundary approximation for N = 1, is a monotone increasing step function; (ii) the uniform convergence of U_Δt and ?U_Δt, to u and ?u, respectively, on D ; (iii) the uniform convergence to the Hölder continuous, monotone increasing free boundary x on [0, T] of the piecewise linear sequence x_Δt, where x_Δt interpolates x _Δt, in one space dimension; (iv) a constructive existence proof for u and x in prescribed regularity classes.     

Consistent interpolation of equidistantly sampled data

Let u₁, u₂, …, u_N with u_n∈? denote the values of a function recorded or computed at N real and equidistant abscissa values t_n=nΔt+t₀ for n=1, …, N. A consistent interpolation operator L , as defined in this paper, interpolates these function values for N new abscissas t_n = (n+½)Δt+t₀, the first N?1 of which are halfway between those originally given while the last one is outside of the original abscissa range. Application of L to these interpolated function values produces the last N?1 samples u₂, u₃, …, u_N of the original data plus one extrapolated function value u_N+1. Hence, L ² is essentially a shift operator, but with a prediction component. The difference between various interpolation methods (e.g. polynomials, Fourier series) is now reduced to the way in which u_N+1 is determined. This concept not only permits a uniform view at interpolation by quite different classes of functions but also allows the creation of more general interpolation, differentiation, and integration formulas, which can be tailored to particular problems. Copyright © 2008 John Wiley & Sons, Ltd.     

A numerical method for computing radially symmetric solutions of a dissipative nonlinear modified Klein‐Gordon equation

In this article we develop a finite‐difference scheme to approximate radially symmetric solutions of a dissipative nonlinear modified Klein‐Gordon equation subject to smooth initial conditions ? and ψ in an open sphere D around the origin, with constant internal and external damping coefficients—β and γ, respectively—, and nonlinear term of the form G′(w) = w^p, with p > 1 an odd number. The functions ? and ψ are radially symmetric in D, and ?, ψ, r?, and rψ are assumed to be small at infinity. We prove that our scheme is consistent order ??(Δt²) + ??(Δr²) for G′ identically equal to zero and provide a necessary condition for it to be stable order n. Part of our study will be devoted to compare the physical effects of β and γ. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Upper functions for lévy processes having only negative jumps

Let X_t,t ≥ 0 be a real valued process with stationary independentincrements having only negative jumps. We obtain b(t) such that lim sup X_t )/b(t) equals a finite positive constant with probability one as t → 0 and t → ∞ under extra condition. The hypotheses about the behavior of Lévy measure near zero and infinity are necessary to guarantee that the lim sup is positive     

On a class of nonlocal parabolic equations

We consider a boundary value problem for parabolic equations with nonlocal nonlinearity of such a form that favorably differs from other equations in that it leads to partial differential equations that have important properties of ordinary differential equations. Local solvability and uniqueness theorems are proved, and an analog of the Painlevé singular nonfixed points theorem is proved. In this case, there is an alternative—either a solution exists for all t ≥ 0 or it goes to infinity in a finite time t = T (blowup mode). Sufficient conditions for the existence of a blowup mode are given.     

Long-time decay of the solutions of the Davey—Stewartson II equations

Summary Using the method of inverse scattering, the sup-norms of the solutions of the Davey—Stewartson II equations are shown to decay in the order of 1/¦t¦ as ¦t¦ goes to infinity. In the focusing case this result is obtained for small initial data, whereas in the defocusing case it is obtained for general initial data.