Phragmen-Lindelöf type results for second order quasilinear parabolic equations in ℝ2

In this paper Phragmen-Lindelöf type growth-decay estimates are derived for solutions of initial-boundary value problems associated with a class of quasilinear parabolic equations defined on a semi-infinite strip in ². The particular problems considered are ones in which homogeneous initial data and homogeneous Dirichlet conditions on the long sides of the strip are prescribed.This research was carried out while the first author held a visiting appointment at Cornell University and was partially supported by NSF grant #DMS-9100876.     

Asymptotic stability for the Navier–Stokes equations in L n

Considering the Navier–Stokes equations in W ì \mathbbRⁿ\Omega \subset {\mathbb{R}}^n, we prove the asymptotic stability for weak solutions in the marginal class u ∈ C _B (0, ∞; L ⁿ ) with arbitrary initial and external perturbations.     

Existence of global L∞ solutions to a generalized n × n hyperbolic system of LeRoux type

In this article, we give the existence of global L^∞ bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of LeRoux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2 × 2 to n × n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v₁=0} is another difficulty. We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.     

On the Feynman–Kac representation for solutions of the Cauchy problem for parabolic equations in divergence form

We consider the Cauchy problem for general second–order uniformly elliptic linear equation in divergence form. We give a stochastic representation of bounded weak solutions of the problem in terms of solutions of associated linear backward stochastic differential equations. Our representation may be considered as an extension of the classical Feynman–Kac formula.     

Asymptotic properties,stability and “near” stationary of parabolic partial differential equations with wide bandwidth inputs

Let u^e(x, t) denote the weak solution to a parabolic PDE whose inputs and coefficients are wide bandwidth stochastic processes, and the bandwidth goes to as e-->0. Under known conditions, u^e(?,?) converges weakly to a certain parabolic PDE driven by a (finite dimensional or cylindrical) Wiener process. A key problem in applications concerns the behavior of u^e(x, t) for large time and small e . This is, perhaps, the main stability problem for such stochastic PDEs. We show (under appropriate conditions) that the tail of u^e is (in the sense of distributions) close to a stationary solution of the limit (Wiener process driven) system, for small e. The methods are purely probabilistic, and depend on interpretations of the solutions u^e(?,?) as functionals of certain diffusions with wide bandwidth coefficients. No PDE methods are used.     

Geometry of unbounded domains,poincaré inequalities and stability in semilinear parabolic equations

We investigate thc close relations existing between certain geometric properties of domains Ω of R^N, the validity of Poincark inequalities in Ω, and the behavior of solutions of semilinear parabolic equations. For the equation u_t-△u=|u|^p-1 we obtain a purely geometric, necessary and sufficient condition on Ω, for the 0 solution to be asymptotically (and exponentially) stable in L^r(ω)1<r<∞ when r is supercritical(r>N(p-1)/2 . The condition is that the inradius of Ω be finite. The result is different for r critical. For the equation u_t-△u=u^p-μ|u|^q,q≥p>1,μ>0 we prove that the finiteness of the inradius is a necessary and sufficient condition for global existence and boundedness of all nonnegative solutions.     

On the boundedness of the L1-norm of Walsh–Fejér kernels

In this paper we establish an iteration for the $L^1$-norm of Walsh–Fejér kernels with the assumption that the Walsh functions are ordered in Paley's sense. We use this iteration to prove some properties of this sequence, including that its supremum is exactly equal to $\frac{17}{15}$.     

On Minimal Projections in l∞(n)

 In this paper we present an estimate of the relative projection constant for a particular class of subspaces of of codimension two. In some cases the exact value of will be calculated. Also Theorem 2.5 from [11] will be generalized.     

Carmichael numbers with a totient of the form a2?+?nb2

Let φ be the Euler function. Fix , and let be an arbitrary set of primes of positive lower natural density. Using a variant of the Alford–Granville–Pomerance construction, we show that there are infinitely many Carmichael numbers N with a totient of the form , where and is a nonempty product of primes from the set . In particular, for any fixed natural number n, there are infinitely many Carmichael numbers N such that φ(N) = a ² + nb ² for some positive integers a and b.     

Generalized Hyers-Ulam stability for general additive functional equations in quasi-β-normed spaces

In 1940 S.M. Ulam proposed the famous Ulam stability problem. In 1941 D.H. Hyers solved the well-known Ulam stability problem for additive mappings subject to the Hyers condition on approximately additive mappings. The first author of this paper investigated the Hyers-Ulam stability of Cauchy and Jensen type additive mappings. In this paper we generalize results obtained for Jensen type mappings and establish new theorems about the Hyers-Ulam stability for general additive functional equations in quasi-β-normed spaces.     

Asymptotic solutions for large time of Hamilton–Jacobi equations in Euclidean n space

We study the large time behavior of solutions of the Cauchy problem for the Hamilton–Jacobi equation ut+H(x,Du)=0$u_t+H(x,Du)=0$ in Rn×(0,∞)${\mathbf{R}}^n\times (0,\infty )$, where H(x,p)$H(x,p)$ is continuous on Rn×Rn${\mathbf{R}}^n\times {\mathbf{R}}^n$ and convex in p  . We establish a general convergence result for viscosity solutions u(x,t)$u(x,t)$ of the Cauchy problem as t→∞$t\to \infty$.     

Cα regularity of weak solutions of non-homogeneous ultraparabolic equations with drift terms

We consider a class of non-homogeneous ultraparabolic differential equations with singular drift terms arising from some physical models, and prove that weak solutions are H?lder continuous, which are sharp in some sense and also generalize the well-known De Giorgi-Nash-Moser theory to degenerate parabolic equations satisfying the H?rmander hypoellipticity condition. The new ingredients are manifested in two aspects: on the one hand, for lower-order terms, we exploit a new Sobolev inequality sui...     

Search for JPC=even++ glueballs in radiative decay of J/Ψ

Based on the general analysis of decay width and branching ratio of two pseudo scalar meson channels, two sets of discriminants between mesons and glueballs for1 = 0,J ^PC = even⁺⁺ unflavored hadrons with the mass between 1.2 and 2.9 GeV are suggested. Known1 = 0,J ^PC = 2⁺⁺, f₂(1525) particle is discriminated as a typical meson. The way to discriminate new1 = 0,J ^PC = even⁺⁺ unflavored hadrons is discussed.     

On ω-limit sets of ordinary differential equations in Banach spaces

Let X be an infinite-dimensional real Banach space. We classify ω-limit sets of autonomous ordinary differential equations x^′=f(x), x(0)=x₀, where f:X→X is Lipschitz, as being of three types I-III. We denote by SX the class of all sets in X which are ω-limit sets of a solution to (1), for some Lipschitz vector field f and some initial condition x₀∈X. We say that S∈SX is of type I if there exists a Lipschitz function f and a solution x such that S=Ω(x) and . We say that S∈SX is of type II if it has non-empty interior. We say that S∈SX is of type III if it has empty interior and for every solution x (of Eq. (1) where f is Lipschitz) such that S=Ω(x) it holds . Our main results are the following: S is a type I set in SX if and only if S is a closed and separable subset of the topological boundary of an open and connected set U⊂X. Suppose that there exists an open separable and connected set U⊂X such that , then S is a type II set in SX. Every separable Banach space with a Schauder basis contains a type III set. Moreover, in all these results we show that in addition f may be chosen Ck-smooth whenever the underlying Banach space is Ck-smooth.     

Global solvability of the Navier–Stokes equations with a free surface in the maximal Lp-Lq regularity class

We consider the motion of incompressible viscous fluids bounded above by a free surface and below by a solid surface in the N-dimensional Euclidean space for $N\ge 2$. The aim of this paper is to show the global solvability of the Navier–Stokes equations with a free surface, describing the above-mentioned motion, in the maximal $L_p\text{-}{L}_q$ regularity class. Our approach is based on the maximal $L_p\text{-}{L}_q$ regularity with exponential stability for the linearized equations, and also it is proved that solutions to the original nonlinear problem are exponentially stable.     

Almost automorphically-forced flows on S1 or R in one-dimensional almost periodic semilinear heat equations

In this paper, we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost periodically-forced scalar reaction-diffusion equation:u_t = u_xx + f(t, u, ux), t > 0, 0 < x < L(0.1)with the periodic boundary condition u(t, 0) = u(t, L), u_x(t, 0) = u_x(t, L),(0.2)where f is uniformly almost periodic in t. In particular, we study the topological structure of the limit sets of the skew-product semiflow. It is ...     

Embedding of Hpω in the class eL

We obtain the necessary conditions for the embedding H _p ^ω ⊂ e ^L (1≤p∞) with convex modulus of continuity ω in terms of this modulus. In the case p=1 these conditions are also sufficient.