The Semilinear Wave Equation on Asymptotically Euclidean Manifolds

We consider the quadratically semilinear wave equation on (? ^d , 𝔤), d ≥ 3. The metric 𝔤 is non-trapping and approaches the Euclidean metric like ?x?^?ρ. Using Mourre estimates and the Kato theory of smoothness, we obtain, for ρ > 0, a Keel–Smith–Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence for the nonlinear problem with small initial data for ρ ≥ 1. Long time existence means that, for all n > 0, the life time of the solution is a least δ^?n , where δ is the size of the initial data in some appropriate Sobolev space. Moreover, for d ≥ 4 and ρ > 1, we obtain global existence for small data.     

Spectral Analysis for Stationary Solutions of the Cahn–Hilliard Equation in ℝ d

We consider the spectrum associated with three types of bounded stationary solutions for the Cahn–Hilliard equation on ? ^d , d ≥ 2: radial solutions, saddle solutions (only for d = 2), and planar periodic solutions. In particular, we establish spectral instability for each type of solution. The important case of multiply periodic solutions does not fit into the framework of our approach, and we do not consider it here.     

Qualitative Properties of Saddle-Shaped Solutions to Bistable Diffusion Equations

We consider the elliptic equation ? Δu = f(u) in the whole ?^2m , where f is of bistable type. It is known that there exists a saddle-shaped solution in ?^2m . This is a solution which changes sign in ?^2m and vanishes only on the Simons cone 𝒞 = {(x ¹, x ²) ∈ ? ^m × ? ^m : |x ¹| = |x ²|}. It is also known that these solutions are unstable in dimensions 2 and 4.

In this article we establish that when 2m = 6 every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution.

These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.     

Reaction-Diffusion Equations with Polynomial Drifts Driven by Fractional Brownian Motions

A reaction-diffusion equation on [0, 1] ^d with the heat conductivity κ > 0, a polynomial drift term and an additive noise, fractional in time with H > 1/2, and colored in space, is considered. We have shown the existence, uniqueness and uniform boundedness of solution with respect to κ. Also we show that if κ tends to infinity, then the corresponding solutions of the equation converge to a process satisfying a stochastic ordinary differential equation.     

The global attractor of a non-local PDE model with delay for population dynamics in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we consider a non-local PDE model with delay for population dynamics in ℝ ⁿ . First, we prove the existence and uniqueness of weak solutions under some suitable decayed assumptions on non-local term at infinity. Then, we obtain the global attractor by proving ω-limit compactness property of the solution operator semigroup.     

Nonexistence of Global Solutions to Nonlinear Stochastic Wave Equations in Mean L p -Norm

This article is concerned with explosive solutions of the initial-boundary problem for a class of nonlinear stochastic wave equations in a domain 𝒟 ? ? ^d . Under appropriate conditions on the initial data, the nonlinear term and the noise intensity, it is proved in Theorem 3.4 that there cannot exist a global solution and the local solution will blow up at a finite time in the mean L ^p  ? norm for p ≥ 1. An example is given to show the application of this theorem.     

On the Centralizers of Derivations with Engel Conditions

Let R be a noncommutative prime ring and d, δ two nonzero derivations of R. If δ([d(x), x] _n ) = 0 for all x ∈ R, then char R = 2, d ² = 0, and δ = αd, where α is in the extended centroid of R. As an application, if char R ≠ 2, then the centralizer of the set {[d(x), x] _n  | x ∈ R} in R coincides with the center of R.     

On the domain dependence of solutions to the Navier–Stokes equations of a two‐dimensional compressible flow

We consider the Navier–Stokes equations for compressible, barotropic flow in two space dimensions, with pressure satisfying p(?)=a?log^d(?) for large ?, here d>1 and a>0. After introducing useful tools from the theory of Orlicz spaces, we prove a compactness result for the solution set of the equations with respect to the variation of the underlying bounded spatial domain. Especially, we get a general existence theorem for the system in question with no restrictions on smoothness of the bounded spatial domain. Copyright © 2009 John Wiley & Sons, Ltd.     

On Kirchhoff's Model of Parabolic Type

In this article, the existence of a global strong solution for all finite time is derived for the Kirchhoff's model of parabolic type. Based on exponential weight function, some new regularity results which reflect the exponential decay property are obtained for the exact solution. For the related dynamics, the existence of a global attractor is shown to hold for the problem when the non-homogeneous forcing function is either independent of time or in L^∞(L²). With the finite element Galerkin method applied in spatial direction keeping time variable continuous, a semidiscrete scheme is analyzed, and it is also established that the semidiscrete system has a global discrete attractor. Optimal error estimates in L^∞(H¹) norm are derived which are valid uniformly in time. Further, based on a backward Euler method, a completely discrete scheme is analyzed and error estimates are derived. It is also further, observed that in cases where f = 0 or f = O(e^?γ₀t) with γ₀ > 0, the discrete solutions and error estimates decay exponentially in time. Finally, some numerical experiments are discussed which confirm our theoretical findings.     

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.     

Existence and non-existence of periodic solutions of generalized Lineard equations with damping of no lower bound

This paper establishes criteria for the existence and non-existence of nonzero periodic solutions of the generalized Liénard equationx +f(x,x)x +g(x)=0. The main goal is to study to what extent the dampingf can be small so as to guarantee the existence of nonzero periodic solutions of such a system. With some standard additional assumptions we prove that if for a small ¦x¦, ^± ¦f(x,y)¦^–1 dy=±, then the system has at least one nonzero periodic solution, otherwise, the system has no nonzero periodic solution. Many classical and well-known results can be proved as corollaries to ours.Supported by the National Natural Science Foundation of China.     

On a Nonlinear Heat Equation Associated with Dirichlet–Robin Conditions

This article is devoted to the study a nonlinear heat equation associated with Dirichlet-Robin conditions. At first, we use the Faedo–Galerkin and the compactness method to prove existence and uniqueness results. Next, we consider the properties of solutions. We obtain that if the initial condition is bounded then so is the solution and we also get asymptotic behavior of solutions as t → + ∞. Finally, we give numerical results.     

Asymptotic behaviour of solutions of quasilinear evolutionary partial differential equations of parabolic type on unbounded spatial intervals

We study the long‐time behaviour of solutions to a quasilinear parabolic problem on a half‐line. The main result lies in showing the existence of a positive solution that converges to the travelling wave of solution to the stationary problem on the whole line. The main tools used here are the zero number theory and the concentration compactness principle. This result is a generalization of a result know for semilinear parabolic equations. Copyright © 2006 John Wiley & Sons, Ltd.     

Extensions of Banach's Contraction Principle

Let A and B be nonempty subsets of a metric space. As a non-self mapping T: A → B does not necessarily have a fixed point, it is of considerable interest to find an element x that is as close to Tx as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x such that the error d(x, Tx) is minimum. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, of the fixed point equation Tx = x when there is no exact solution. As d(x, Tx) is at least d(A, B), a best proximity point theorem achieves an absolute minimum of the error d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). This article furnishes extensions of Banach's contraction principle to the case of non-self mappings. On account of the preceding argument, the proposed generalizations are formulated as best proximity point theorems for non-self contractions.     

Algebraic Versus Homological Equivalence for Singular Varieties

Let Y ? ?^N be a possibly singular projective variety, defined over the field of complex numbers. Let X be the intersection of Y with h general hypersurfaces of sufficiently large degrees. Let d > 0 be an integer, and assume that dimY = n + h and dimY_sing ≤ min {d + h ? 1, n ? 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H_2(d+h)(Y; ?) is nonzero. In the present article, we prove that the restriction of Z to X is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case Y is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.     

Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs via Stochastic Calculus

We consider two quasi-linear initial-value Cauchy problems on ? ^d : a parabolic system and an hyperbolic one. They both have a first order non-linearity of the form φ(t, x, u)·?u, a forcing term h(t, x, u) and an initial condition u ₀ ∈ L ^∞(? ^d ) ∩ C ^∞(? ^d ), where φ (resp. h) is smooth and locally (resp. globally) Lipschitz in u uniformly in (t, x). We prove the existence of a unique global strong solution for the parabolic system. We show the existence of a unique local strong solution for the hyperbolic one and we give a lower bound regarding its blow up time. In both cases, we do not use weak solution theory but a direct construction based on parabolic schemes studied via a stochastic approach and a regularity result for sequences of parabolic operators. The result on the hyperbolic problem is performed by means of a non-classical vanishing viscosity method.     

Hölder Flow and Differentiability for SDEs with Nonregular Drift

We prove the existence of a stochastic flow of Hölder homeomorphisms for solutions of SDEs with singular time dependent drift having only certain integrability properties. We also show that the solution map x → X ^x is differentiable in a weak sense.     

Singly Periodic Solutions of the Allen-Cahn Equation and the Toda Lattice

The Allen-Cahn equation ? Δu = u ? u ³ in ?² has family of trivial singly periodic solutions that come from the one dimensional periodic solutions of the problem ?u″ =u ? u ³. In this paper we construct a non-trivial family of singly periodic solutions to the Allen-Cahn equation. Our construction relies on the connection between this equation and the infinite Toda lattice. We show that for each one-soliton solution to the infinite Toda lattice we can find a singly periodic solution to the Allen-Cahn equation, such that its level set is close to the scaled one-soliton. The solutions we construct are analogues of the family of Riemann minimal surfaces in ?³.     

A note on some piecewise-linear difference equations with Mersenne-type periodic solutions

This paper studies solutions of some piecewise-linear difference equations. In two particular cases, a descent argument is used to show that all solutions are periodic with either prime period 3(2 ^k  ? 1) or 6(2 ^k  ? 1) for some k ≥ 1. The existence of solutions with such periods is also considered.