Local bifurcation-branching analysis of global and ��blow-up�� patterns for a fourth-order thin film equation

Countable families of global-in-time and blow-up similarity sign-changing patterns of the Cauchy problem for the fourth-order thin film equation (TFE-4)

Stabilization of Solutions of the Nonlinear Fokker–Planck Equation

One considers the equation $$ \mathrm{div}\left( {{u^{\sigma }}Du} \right)+b(x)Du-{u_t}=f(x)g(u),\quad x\in {{\mathbb{R}}^n},\quad t\in \left( {0,\infty } \right), $$ where $ b:{{\mathbb{R}}^n}\to {{\mathbb{R}}^n} $ and $ f:{{\mathbb{R}}^n}\to [0,\infty ) $ are locally bounded measurable functions, g: (0,∞)?→?(0,∞) is continuous and nondecreasing, One obtains the conditions ensuring that its positive solutions stabilize to zero as t?→?∞.     

On the Number of Topological Types Occurring in a Parameterized Family of Arrangements

Let be an o-minimal structure over ℝ, a closed definable set, and

Exotic smooth structures on small 4-manifolds

Let M be either or . We construct the first example of a simply-connected irreducible symplectic 4-manifold that is homeomorphic but not diffeomorphic to M. Dedicated to Ronald J. Stern on the occasion of his sixtieth birthday Mathematics Subject Classification (2000)  Primary 57R55; Secondary 57R17, 57M05     

Bound states for a coupled Schrödinger system

We consider the existence of bound states for the coupled elliptic system

Automorphisms of a finitary factor power of an infinite symmetric group

We consider a semigroup FP_\textfin⁺ ( \mathfrakS_\textfin( \mathbbN ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) defined as a finitary factor power of a finitary symmetric group of countable order. It is proved that all automorphisms of FP_\textfin⁺ ( \mathfrakS_\textfin( \mathbbN ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) are induced by permutations from \mathfrakS( \mathbbN ) \mathfrak{S}\left( \mathbb{N} \right) .     

The q-Variation of Functions and Spectral Integration from Dominated Ergodic Estimates

For $1\leq q < \infty$, let $\mathfrak{M}_{q}\left( \mathbb{T}\right)$, (respectively, $\mathfrak{M}_{q}\left( \mathbb{R}\right) $) denote the Banach algebra consisting of the bounded complex-valued functions having uniformly bounded $q$-variation on the dyadic arcs of the unit circle, (respectively, on the dyadic intervals of the real line). Suppose that $(\Omega,\mu)$ is a $\sigma$-finite measure space, $1< p < \infty$, and $T:L^{p}(\mu)\rightarrow L^{p}(\mu)$ is a bounded, invertible, separation-preserving linear operator such that the two-sided ergodic means of the linear modulus of $T$ are uniformly bounded in norm. We show that there is a real number $q_{_{0}} > 1$ such that whenever $1\leq q < q_{_{0}}$, $T $ has a norm-continuous functional calculus associated with $\mathfrak{M}_{q}\left(\mathbb{T}\right) $. Our approach is rooted in a dominated ergodic theorem of Mart\{\i}n--Reyes and de la Torre which assigns $T$ a canonical family of bilateral $A_{p}$ weight sequences. We first establish the relevant multiplier properties of $\mathfrak{M}_{q}\left( \mathbb{R}\right) $ classes in weighted settings, transfer the outcome to $\mathfrak{M}_{q}\left(\mathbb{T}\right) $, and then apply the consequent $\mathfrak{M}_{q}\left(\mathbb{T}\right) $ multiplier theorem for weighted settings to the spectral decomposition of $T$. The desired $\mathfrak{M}_{q}\left(\mathbb{T}\right)$-functional calculus for $T$ then results from an extension criterion for spectral integration obtained in the general Banach space setting. The multiplier result for $\mathfrak{M}_{q}\left( \mathbb{R}\right) $ shown at the outset of this process expands the scope of the weighted Marcinkiewicz multiplier theorem from $q=1$ to appropriate values of $q > 1$     

Martin Boundary of a Killed Random Walk on a Half-Space

A complete representation of the Martin boundary of killed random walks on a half-space ℤ ^d−1×ℕ^* is obtained. In particular, it is proved that the corresponding Martin boundary is homemorphic to the half-sphere . The method is based on a combination of ratio limits theorems and large deviation techniques.     

Nonradial solutions for a conformally invariant fourth order equation in \mathbb {R}^4

We consider the following Liouville equation in

Precise asymptotics in the law of the iterated logarithm*

Let X₁, X₂, ... be i.i.d. random variables with EX₁ = 0 and positive, finite variance σ², and set S_n = X₁ + ... + X_n. For any α > −1, β > −1/2 and for κ_n(ε) a function of ε and n such that κ_n(ε) log log n → λ as n ↑ ∞ and , we prove that

Remarks on the Extremal Functions for the Moser-Trudinger Inequality

We will show in this paper that if A is very close to 1, then I（M,λ,m） =supu∈H0^1,n（m）,∫m｜△↓u｜^ndV=1∫Ω（e^αn｜u｜^n/（n-1）-λm∑k=1｜αnun/（n-1）｜k/k!）dV can be attained, where M is a compact-manifold with boundary. This result gives a counter-example to the conjecture of de Figueiredo and Ruf in their paper titled ＂On an inequality by Trudinger and Moser and related elliptic equations＂ （Comm. Pure. Appl. Math., 55, 135-152, 2002）.     

Constant Angle Property and Canonical Principal Directions for Surfaces in {\mathbb{M}^{2}(c) \times {\mathbb{R}_{1}}}

In this paper, we study surfaces in Lorentzian product spaces ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ . We classify constant angle spacelike and timelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ . Moreover, complete classifications of spacelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ and timelike surfaces in ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ with a canonical principal direction are obtained. Finally, a new characterization of the catenoid of the 3rd kind is established, as the only minimal timelike surface with a canonical principal direction in Minkowski 3–space.     

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

This paper is concerned with a nonlocal hyperbolic system as follows utt = △u ＋ （∫Ωvdx ）^p for x∈R^N,t〉0 ,utt = △u ＋ （∫Ωvdx ）^q for x∈R^N,t〉0 ,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.     

Infinite-Time Quenching in a Fast Diffusion Equation with Strong Absorption

This work is concerned with the fast diffusion equation , where 0 < m < 1 and κ < 1. A global positive solution is said to quench regularly in infinite time if for some bounded sequence and some , and if for all compact . It is shown that such regular quenching in infinite time occurs for a large class of initial data if κ > m , whereas it is impossible in one space dimension when κ < −m and the solution is radially symmetric and nondecreasing for x > 0.      

Almost Periodicity of Parabolic Evolution Equations with Inhomogeneous Boundary Values

We show the existence and uniqueness of the (asymptotically) almost periodic solution to parabolic evolution equations with inhomogeneous boundary values on \mathbbR{\mathbb{R}} and \mathbbR_±\mathbb{R}_{\pm}, if the data are (asymptotically) almost periodic. We assume that the underlying homogeneous problem satisfies the ‘Acquistapace–Terreni’ conditions and has an exponential dichotomy. If there is an exponential dichotomy only on half intervals ( − ∞, − T] and [T, ∞), then we obtain a Fredholm alternative of the equation on \mathbbR{\mathbb{R}} in the space of functions being asymptotically almost periodic on \mathbbR₊{\mathbb{R}}_{+} and \mathbbR_-\mathbb{R}_{-}.     

Approximation of bandlimited functions by finite exponential sums

For a compact set K in ℝ ⁿ , let B ² _K be the set of all functions f ∈ L ²(ℝ²) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B ² _K by finite exponential sums