H m -modified wave operator for nonlinear Hartree equation in space of dimensionn≥2

We consider the scattering problem for the Hartree equation with potential |x|⁻¹ in a space of dimensionn≥2. We prove the existence ofH ^m -modified wave operator for Hartree equation on a dense set of a neighborhood of zero inH ^m (ℝ ⁿ ), meanwhile, we obtain also the global existence for the Cauchy problem of Hartree equation in a space of dimensionn≥2. This project is supported by the National Natural Science Foundation of China, 19601005     

On estimates of biharmonic functions on Lipschitz and convex domains

Using Maz ’ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in ℝⁿ. Forn ≥ 8, combinedwitharesultin[18], these estimates lead to the solvability of the L^p Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of p. In the case of convex domains, the estimates allow us to show that the L^p Dirichlet problem is uniquely solvable for any 2 − ε < p < ∞ and n ≥ 4.     

Modified Busemann-Petty problem on sections of convex bodies

The Busemann-Petty problem asks whether convex origin-symmetric bodies in ℝ ⁿ with smaller central hyperplane sections necessarily have smallern-dimensional volume. It is known that the answer is affirmative ifn≤4 and negative ifn≥5. In this article we replace the assumptions of the original Busemann-Petty problem by certain conditions on the volumes of central hyperplane sections so that the answer becomes affirmative in all dimensions. The first-named author was supported in part by the NSF grant DMS-0136022 and by a grant from the University of Missouri Research Board.     

Local and Global Existence of Solutions to Initial Value Problems of Modified Nonlinear Kawahara Equations

This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third ＋α the second partial dervative of u to x ,the second the third ＋β the third partial dervative of u to x ,the second the thire ＋γ the fifth partial dervative of u to x = 0,（x,t）∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo（x） ∈ H^s（R） with s ≥ 1/4, and a global solution exists if s ≥ 2.     

Maximum principles for the polyharmonic equation on Lipschitz domains

The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in ℝ³. In ℝn,n≥4, the Agmon-Miranda maximum principle andL ^p-Dirichlet estimates for certainp>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2m+1 theL ^p Dirichlet problem for Δ ^m fails to be solvable forp>2(n−1)/(n−3). Supported in part by the NSF.     

<Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Bold">1,</Emphasis><Emphasis Type="Italic">α</Emphasis></Superscript> regularity for infinity harmonic functions in two dimensions

We propose a new method for showing C ^{1, α} regularity for solutions of the infinity Laplacian equation and provide full details of the proof in two dimensions. The proof for dimensions n ≥ 3 depends upon some conjectured local gradient estimates for solutions of certain transformed PDE. LCE is supported in part by NSF Grant DMS-0500452. OS was supported in part by the Miller Institute for Basic Research in Science, Berkeley.     

Existence and non-existence of global solutions for a semilinear heat equation

The existence and non-existence of global solutions and theL ^p blow-up of non-global solutions to the initial value problemu′(t)=Δu(t)+u(t) ^γ onR ⁿ are studied. We consider onlyγ>1. In the casen(γ − 1)/2=1, we present a simple proof that there are no non-trivial global non-negative solutions. Ifn(γ−1)/2≦1, we show under mild technical restrictions that non-negativeL ^p solutions always blow-up inL ^p norm in finite time. In the casen(γ−1)/2>1, we give new sufficient conditions on the initial data which guarantee the existence of global solutions. Research partially supported by NSF grant MCS79-03636.     

Nonnegative solutions of an elliptic equation with a singular potential

We study the behavior of nonnegative solutions of the Dirichlet problem for a linear elliptic equation with a singular potential in the ball B = B(0,R) ⊂ R ⁿ (n ≥ 3), R ≤ 1. We find an exact condition on the potential ensuring the existence or absence of a nonnegative solution of that problem.     

The existence of global attractors for 2D Navier-Stokes equations in <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> spaces

In this paper, we prove that the 2D Navier-Stokes equations possess a global attractor in Hk（Ω,R2） for any k ≥ 1, which attracts any bounded set of Hk（Ω,R2） in the H^k-norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor. This extends Ma, Wang and Zhong＇s conclusion.     

H-designs with the properties of resolvability or (1, 2)-resolvability

An H-design is said to be (1, α)-resolvable, if its block set can be partitioned into α-parallel classes, each of which contains every point of the design exactly α times. When α = 1, a (1, α)-resolvable H-design of type g ⁿ is simply called a resolvable H-design and denoted by RH(g ⁿ ), for which the general existence problem has been determined leaving mainly the case of g ≡ 0 (mod 12) open. When α = 2, a (1, 2)-RH(1 ⁿ ) is usually called a (1, 2)-resolvable Steiner quadruple system of order n, for which the existence problem is far from complete. In this paper, we consider these two outstanding problems. First, we prove that an RH(12 ⁿ ) exists for all n ≥ 4 with a small number of possible exceptions. Next, we give a near complete solution to the existence problem of (1, 2)-resolvable H-designs with group size 2. As a consequence, we obtain a near complete solution to the above two open problems.     

Estimates on Conjectures of Minkowski and woods

Let ℝ ⁿ be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in ℝ ⁿ of radius $ \sqrt {n/4} $ \sqrt {n/4} contains a point of ∧. This is known to be true for n ≤ 8. Here we give estimates on a more general conjecture of Woods for n ≥ 9. This leads to an improvement for 9 ≤ n ≤ 22 on estimates of Il’in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms.     

On Small World Semiplanes with Generalised Schubert Cells

The well known “real-life examples” of small world graphs, including the graph of binary relation: “two persons on the earth know each other” contains cliques, so they have cycles of order 3 and 4. Some problems of Computer Science require explicit construction of regular algebraic graphs with small diameter but without small cycles. The well known examples here are generalised polygons, which are small world algebraic graphs i.e. graphs with the diameter d≤clog  _k−1(v), where v is order, k is the degree and c is the independent constant, semiplanes (regular bipartite graphs without cycles of order 4); graphs that can be homomorphically mapped onto the ordinary polygons. The problem of the existence of regular graphs satisfying these conditions with the degree ≥k and the diameter ≥d for each pair k≥3 and d≥3 is addressed in the paper. This problem is positively solved via the explicit construction. Generalised Schubert cells are defined in the spirit of Gelfand-Macpherson theorem for the Grassmanian. Constructed graph, induced on the generalised largest Schubert cells, is isomorphic to the well-known Wenger’s graph. We prove that the family of edge-transitive q-regular Wenger graphs of order 2q ⁿ , where integer n≥2 and q is prime power, q≥n, q>2 is a family of small world semiplanes. We observe the applications of some classes of small world graphs without small cycles to Cryptography and Coding Theory.     

Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions

In this paper, the author considers the Cauchy problem for semilinear wave equations with critical exponent in n≥4 space dimensions. Under some positivity conditions on the initial data, it is proved that there can be no global solutions no matter how small the initial data are.     

Algebraic families of smooth hyperbolic surfaces of low degree in ℙ ℂ 3

We reprove (after a paper of Y.T. Siu appeared in 1987) a simple vanishing theorem for the Wronskian of Brody curves under a suitable assumption on the existence of global meromorphic connections. Next we give a slight improvement of a result due to Y.T, Siu and A.M. Nadel (Duke Math. J., 1989) on the algebraic degeneracy of entire holomorphic curves contained in certain hypersurfaces of ℙ _ℂ ⁿ . Especially, their result is generalized to a larger class of hypersurfaces. Our method produces algebraic families of smooth hyperbolic surfaces in ℙ _ℂ ³ for all degreesd≥14; this brings us somewhat nearer than previously known from the expected ranged≥5.     

A fourth‐order parabolic equation in two space dimensions

In this paper, we consider an initial‐boundary problem for a fourth‐order nonlinear parabolic equations. The problem as a model arises in epitaxial growth of nanoscale thin films. Based on the L^p type estimates and Schauder type estimates, we prove the global existence of classical solutions for the problem in two space dimensions. Copyright © 2007 John Wiley & Sons, Ltd.     

Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities

This paper deals with a coupled system of fourth-order parabolic inequalities |u|t ≥ 2u + |v|q,|v|t ≥ 2v + |u|p in S = Rn × R+ with p,q > 1,n ≥ 1.A FujitaLiouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n4 ≤ max(ppq+11,pqq+11).Since the general maximum-comparison principle does not hold for the fourth-order problem,the authors use the test function method to get the global non-existence of nontrivial solutions.     

On the existence and nonexistence of global solutions for the porous medium equation with strongly nonlinear sources in a cone

In this paper, we study the initial-boundary value problem of the porous medium equation u _t  = Δu ^m  + V(x)u ^p in a cone D = (0, ∞) × Ω, where V(x) ~ (1 + |x|) ^σ . Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω and let l denote the positive root of l ² + (n − 2)l = ω ₁. We prove that if m ≤ p ≤ m + (2 + σ)/(n + l), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if p > m + (2 + σ)/n, then the problem has global solutions for some u ₀ ≥ 0.     

Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation

We prove a statement on the averaging of a hyperbolic initial-boundary-value problem in which the coefficient of the Laplace operator depends on the space L ²-norm of the gradient of the solution. The existence of the solution of this problem was studied by Pokhozhaev. In a space domain in ℝⁿ, n ≥ 3, we consider an arbitrary perforation whose asymptotic behavior in a sense of capacities is described by the Cioranesku-Murat hypothesis. The possibility of averaging is proved under the assumption of certain additional smoothness of the solutions of the limiting hyperbolic problem with a certain stationary capacitory potential. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 236–249, February, 2006.     

Pullback attractors for non-autonomous reaction-diffusion equations on ℝ n

We study the long time behavior of solutions of the non-autonomous reaction-diffusion equation defined on the entire space ℝ ⁿ when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L ²(ℝ ⁿ ) and H ¹(ℝ ⁿ ), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains.