Asymptotic properties of matrix differential operators

For given matrices A(s) and B(s) whose entries are polynomials in s, the validity of the following implication is investigated: ?_ylim_{t → ∞}A(D) y(t) = 0 ? lim_{t → ∞}B(D) y(t) = 0. Here D denotes the differentiation operator and y stands for a sufficiently smooth vector valued function. Necessary and sufficient conditions on A(s) and B(s) for this implication to be true are given. A similar result is obtained in connection with an implication of the form ?_yA(D) y(t) = 0, lim_{t → ∞}B(D) y(t) = 0, C(D) y(t) is bounded ? lim_{t → ∞}E(D) y(t) = 0.     

On the signless Laplacian spectral characterization of the line graphs of T-shape trees

A graph is determined by its signless Laplacian spectrum if no other nonisomorphic graph has the same signless Laplacian spectrum (simply G is DQS). Let T (a, b, c) denote the T-shape tree obtained by identifying the end vertices of three paths P _a+2, P _b+2 and P _c+2. We prove that its all line graphs L(T(a, b, c)) except L(T(t, t, 2t+1)) (t ? 1) are DQS, and determine the graphs which have the same signless Laplacian spectrum as L(T(t, t, 2t + 1)). Let µ₁(G) be the maximum signless Laplacian eigenvalue of the graph G. We give the limit of µ₁(L(T(a, b, c))), too.     

Asymptotic behaviour of a class of nonlinear functional differential equations of third order

This paper deals with nonoscillatory behaviour of solutions of third-order nonlinear functional differential equations of the form y‴ + p(t)y′ + q(t)F(y(g(t))) = 0. It has been shown that under certain conditions on coefficient functions, the nonoscillatory solutions of this equation tends to either zero or ∓∞ as t → ∞.     

Decay of solutions of some nonlinear parabolic inequalities associated with time-dependent convexes

A simple result concerning integral inequalities enables us to give an alternative proof of Waltman's theorem: lim_{t → ∞} ∝^t₀a(s) ds = ∞ implies oscillation of the second order nonlinear equation y″(t) + a(t) f(y(t)) = 0; to prove an analog of Wintner's theorem that relates the nonoscillation of the second order nonlinear equations to the existence of solutions of some integral equations, assuming that lim_{t → ∞} ∝^t₀a(s) ds exists; and to give an alternative proof and to extend a result of Butler. An often used condition on the coefficient a(t) is given a more familiar equivalent form and an oscillation criterion involving this condition is established.     

Stationary distributions of the simplest dynamical systems in Rn perturbed by generalized Poisson process

Let z(t) ∈ Rⁿ be a generalized Poisson process with parameter λ and let A: Rⁿ → Rⁿ be a linear operator. The conditions of existence and limiting properties as λ → ∞ or as λ → 0 of the stationary distribution of the process x(t) ∈ Rⁿ which satisfies the equation dx(t) = Ax(t)dt + dz(t) are investigated.     

Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)^1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0⁺. Then we find, as t → ∞, (P{Z(t)> 0})^αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u^α)^?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.     

On the structure of a semigroup of operators with finite-dimensional ranges

In the present paper, we describe the structure of a strongly continuous operator semigroup T(t) (where T: ?₊ → End X and X is a complex Banach space) for which ImT(t) is a finite-dimensional space for all t > 0. It is proved that such a semigroup is always the direct sum of a zero semigroup and a semigroup acting in a finite-dimensional space. As examples of applications, we discuss differential equations containing linear relations, orbits of a special form, and the possibility of embedding an operator in a C ₀-semigroup.     

On the renewal measure for Gaussian sequences

A form for U(t), the expected number of times a Gaussian sequence falls below a level of t, is given in terms of the mean M(x) and the variance V²(x) functions. It is shown that under general conditions U(t) ∼ M⁽⁻¹⁾(t), t → ∞. Moreover, if M and V are regularly varying at infinity functions, then U(t) − M⁽⁻¹⁾(t) is also regularly varying at infinity. A renewal theorem for stationary Gaussian sequences is given, where it is shown that the asymptotic behavior of U(t) − t/μ is determined by the asymptotic behavior of V²(t)/t.     

Conditions for the convergence in distribution of maxima of stationary normal processes

The asymptotic distribution of the maximum M_n=max_1?t?nξ_t in a stationary normal sequence ξ₁,ξ,… depends on the correlation r_t between ξ₀ and ξ_t. It is well known that if r_t log t → 0 as t → ∞ or if Σr²_t<∞, then the limiting distribution is the same as for a sequence of independent normal variables. Here it is shown that this also follows from a weaker condition, which only puts a restriction on the number of t-values for which rt log t islarge. The condition gives some insight into what is essential for this asymptotic behaviour of maxima. Similar results are obtained for a stationary normal process in continuous time.     

Periodic forcing of solutions of a boundary value problem for a second-order differential equation in Hilbert space

We show that if u is a bounded solution on $\text{R}$⁺ of u″(t) ?Au(t) + f(t), where A is a maximal monotone operator on a real Hilbert space H and f∈L_loc²($\text{R}$⁺;H) is periodic, then there exists a periodic solution ω of the differential equation such that u(t) ? ω(t)   0 and u′(t) ? ω′(t) → 0 as t → ∞. We also show that the two-point boundary value problem for this equation has a unique solution for boundary values in $\text{D(A)}$ and that a smoothing effect takes place.     

The linear model revisited

Following Gerber (1982), the annual gain of an insurance concern, G_t, is modeled as an autoregressive moving average process. The surplus process is defined as U_t = u + Σ^t_k=₁G_k. The distribution of U_t is obtained. It is proved that the traditional limit theorems, i.e., the central limit theorem, the strong law of large numbers and the law of the iterated logarithm, hold for U_t as t → ∞. Under certain conditions, bounds are provided for the probability of non-ruin in a finite time interval. It is proved that, if E[G_t] > 0 as t → ∞, the probability of non-ruin in an infinite time interval is positive.     

Global heat kernel estimates for fractional Laplacians in unbounded open sets

In this paper, we derive global sharp heat kernel estimates for symmetric ??-stable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unbounded C ^1,1 open sets in ${\mathbb R^d}$ : half-space-like open sets and exterior open sets. These open sets can be disconnected. We focus in particular on explicit estimates for p _D (t, x, y) for all t?>?0 and ${x, y\,{\in}\,D}$ . Our approach is based on the idea that for x and y in D far from the boundary and t sufficiently large, we can compare p _D (t, x, y) to the heat kernel in a well understood open set: either a half-space or ${\mathbb R^d}$ ; while for the general case we can reduce them to the above case by pushing x and y inside away from the boundary. As a consequence, sharp Green functions estimates are obtained for the Dirichlet fractional Laplacian in these two types of open sets. Global sharp heat kernel estimates and Green function estimates are also obtained for censored stable processes (or equivalently, for regional fractional Laplacian) in exterior open sets.     

Nonlinear kinetic theory of vehicular traffic

We prove the existence and uniqueness of a continuous solution F = φ + w of the initial-value problem for vehicular traffic according to the nonlinear Prigogine-Herman model, where φ is a suitable t- and x-independent car distribution.We then show that the perturbation w is strongly continuous and strongly differentiable any number of times with respect to the probability of not passing q. Moreover, the derivatives $\text{?}{}^{\mathrm{m}}\text{w}\text{?q}{}^{\mathrm{m}}$ (in the strong sense) satisfy linear systems.We finally investigate the behavior of w(t) as t → + ∞ and, under the assumption that the probability of not passing remains unchanged after the instant t = 0, we prove that lim ∥w(t)∥ = 0 as t → + ∞.     

An abstract approach to Loewner chains

We present a new geometric construction of Loewner chains in one and several complex variables which holds on complete hyperbolic complex manifolds and prove that there is essentially a one-to-one correspondence between evolution families of order d and Loewner chains of the same order. As a consequence, we obtain a univalent solution (f _t : M → N) of any Loewner-Kufarev PDE. The problem of finding solutions given by univalent mappings (f _t : M → ? ⁿ ) is reduced to investigating whether the complex manifold ∪ _t≥0 f _t (M) is biholomorphic to a domain in ? ⁿ . We apply such results to the study of univalent mappings f: B ⁿ → ? ⁿ .     

Single point blow-up for a semilinear initial value problem

The initial value problem on [?R, R] is considered: u_t(t, x) = u_xx(t, x) + u(t, x)^γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥_∞ → ∞ as t → T. For the specific initial value ? = kψ, where ψ ? 0, ψ_xx + ψ^γ = 0, ψ(±R) = 0, k is sufficiently large, it is shown that if x ≠ 0, then lim_{t → T}u(t, x) and lim_{t → T}u_x(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0.     

On the generalized Burgers equation

The present paper is concerned with the initial boundary value problem for the generalized Burgers equation u _t + g(t, u)u _x + f(t, u) = εu _xx which arises in many applications. We formulate a condition guaranteeing the a priori estimate of max |u _x | independent of ε and t and give an example demonstrating the optimality of this condition. Based on this estimate we prove the global existence of a unique classical solution of the problem and investigate the behavior of this solution for ε → 0 and t → + ∞. The Cauchy problem for this equation is considered as well.     

Existence and asymptotic behavior of a functional differential equation in Banach space

Existence and asymptotic behavior of solutions are given for the equation u′(t) = ?A(t)u(t) + F(t,u_t) (t ? 0) and u₀ = ? ? C([?r,0]; X)  C. The space X is a Banach space; the family $\text{{A(t) ¦ 0 ? t ? T}}$ of unbounded linear operators defined on D(A) ? X → X generates a linear evolution system and F: C → X is continuous with respect to a fractional power of A(t₀) for some t₀ ? [0, T].     

On the trace of symmetric stable processes on Lipschitz domains

It is shown that the second term in the asymptotic expansion as t→0 of the trace of the semigroup of symmetric stable processes (fractional powers of the Laplacian) of order α, for any 0<α<2, in Lipschitz domains is given by the surface area of the boundary of the domain. This brings the asymptotics for the trace of stable processes in domains of Euclidean space on par with those of Brownian motion (the Laplacian), as far as boundary smoothness is concerned.     

Asymptotic behavior of trajectories for some nonautonomous,almost periodic processes

In this paper, asymptotics are studied for some almost periodic processes on a complete metric space (X, d): (1) It is shown that any precompact positive trajectory of a contractive periodic process is asymptotically almost periodic as t → +∞. This property does not hold for general almost periodic contractive processes. (2) A compactness result is obtained for weakly almost periodic complete trajectories of some (possibly nonlinear) processes in a uniformly convex Banach space. (3) The existence of almost periodic trajectories is studied for “affine” processes in a uniformly convex Banach space. These results are applicable to some evolution equations of the form $\text{du}\text{dt}\text{+ A (t) u(t) ? f(t)}$, where $\text{?(t)}$ is almost periodic: R → V uniformly convex Banach space and A(t) is a periodic, time-dependent, m-accretive operator in V.     

Spatially monotone homoclinic orbits in nonlinear parabolic equations of super-fast diffusion type

This work deals with positive classical solutions of the degenerate parabolic equation $$u_t=u^p u_{xx} \quad \quad (\star)$$ when p > 2, which via the substitution v = u ^1?p transforms into the super-fast diffusion equation ${v_t=(v^{m-1}v_x)_x}$ with ${m=-\frac{1}{p-1} \in (-1,0)}$ . It is shown that ( ${\star}$ ) possesses some entire positive classical solutions, defined for all ${t \in \mathbb {R}}$ and ${x \in \mathbb {R}}$ , which connect the trivial equilibrium to itself in the sense that u(x, t) → 0 both as t → ?∞ and as t → + ∞, locally uniformly with respect to ${x \in \mathbb {R}}$ . Moreover, these solutions have quite a simple structure in that they are monotone increasing in space. The approach is based on the construction of two types of wave-like solutions, one of them being used for ?∞ < t ≤  0 and the other one for 0 < t <  + ∞. Both types exhibit wave speeds that vary with time and tend to zero as t → ?∞ and t → + ∞, respectively. The solutions thereby obtained decay as x → ?∞, uniformly with respect to ${t \in \mathbb {R}}$ , but they are unbounded as x → + ∞. It is finally shown that within the class of functions having such a behavior as x → ?∞, there does not exist any bounded homoclinic orbit.