Blow‐up estimates for a quasi‐linear reaction–diffusion system

In this paper, some sufficient conditions under which the quasilinear elliptic system ‐div(∣?_u∣^p‐2?_u) = uv, ‐div(∣?_u∣^q‐2?_u) = uv in ?^N(N≥3) has no radially symmetric positive solution is derived. Then by using this non‐existence result, blow‐up estimates for a class of quasilinear reaction–diffusion systems u_t = div (∣?_u∣^p‐2?_u)+uv,vt = div(∣?_v∣^q‐2?_v) +uv with the homogeneous Dirichlet boundary value conditions are obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

Global and blow‐up solutions for non‐linear degenerate parabolic systems

In this paper the degenerate parabolic system u_t=u(u_xx+av). vt=v(v_xx+bu) with Dirichlet boundary condition is studied. For , the global existence and the asymptotic behaviour (α₁=α₂) of solution are analysed. For , the blow‐up time, blow‐up rate and blow‐up set of blow‐up solution are estimated and the asymptotic behaviour of solution near the blow‐up time is discussed by using the ‘energy’ method. Copyright © 2003 John Wiley & Sons, Ltd.     

Semilinear wave equation with time dependent potential

We consider the following semilinear wave equation: (1) for (t,x) ∈ ?_t × ?. We prove that if the potential V(t,x) is a measurable function that satisfies the following decay assumption: ∣V(t,x)∣?C(1+t)(1+∣x∣) for a.e. (t,x) ∈ ?_t × ? where C, σ₀>0 are real constants, then for any real number λ that satisfies there exists a real number ρ(f,g,λ)>0 such that the equation has a global solution provided that 0<ρ?ρ(f,g,λ). Copyright © 2004 John Wiley & Sons, Ltd.     

Several Dimensional θ-Summability and Hardy Spaces

The d-dimensional Hardy spaces H_p ( T × … × T ) (d = d₁ + … + d_kand a general summability method of Fourier series and Fourier transforms are introduced with the help of integrable functions θ_j having integrable Fourier transforms. Under some conditions on θ_j we show that the maximal operator of the θ-means of a distribution is bounded from H_p ( T × … × T ) to L_p ( T ^d) where p₀ < p < ∞ and p₀ < 1 is depending only on the functions θ_j. By an interpolation theorem we get that the maximal operator is also of weak type ( L₁) (i = 1, …, k) where the Hardy space is defined by a hybrid maximal function and if k = 1. As a consequence we obtain that the θ-means of a function (log L)^k–1 converge a.e. to the function in question. If k = 1 then we get this convergence result for all f ∈ L₁. Moreover, we prove that the θ-means are uniformly bounded on the spaces H_p ( T × … × T ) whenever p₀ <p < ∞, thus the θ-means converge to f in ( T × … × T ) norm. The same results are proved for the conjugate θ-means and for d-dimensional Fourier transforms, too. Some special cases of the θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, Riemann, de La Vallée-Poussin, Rogosinski and Riesz summations.     

Function spaces of varying smoothness I

This paper deals with function spaces of varying smoothness. It is a modified version of corresponding parts of [8]. Corresponding spaces of positive smoothness s (x) will be considered in part II. We define the spaces B_p (?ⁿ ), where the function ??: x ? s (x) is negative and determines the smoothness pointwise. First we prove basic properties and then we use different wavelet decompositions to get information about the local smoothness behavior. The main results are characterizations of the spaces B_p (?ⁿ ) by weighted sequence space norms of the wavelet coefficients. These assertions are used to prove an interesting connection to the so‐called two‐microlocal spaces C^{s,s ′} (x⁰). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finding blocks and other patterns in a random coloring of ℤ

Let ξ = (ξ_k)_k∈? be i.i.d. with P(ξ_k = 0) = P(ξ_k = 1) = 1/2, and let S: = (S_k) be a symmetric random walk with holding on ?, independent of ξ. We consider the scenery ξ observed along the random walk path S, namely, the process (χ_k := ξ). With high probability, we reconstruct the color and the length of blockⁿ, a block in ξ of length ≥ n close to the origin, given only the observations (χ_k). We find stopping times that stop the random walker with high probability at particular places of the scenery, namely on blockⁿ and in the interval [?3ⁿ,3ⁿ]. Moreover, we reconstruct with high probability a piece of ξ of length of the order 3 around blockⁿ, given only 3 observations collected by the random walker starting on the boundary of blockⁿ. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Construction of Z-cyclic triple whist tournaments

Let p = 2^kt + 1 be a prime where t>1 is an odd integer, k ≥ 2. Methods of constructing a Z-cyclic triple whist tournament TWh(p) are given. By such methods we construct a Z-cyclic TWh(p) for all primes p,p≡1(mod 4), 29 ≤ p ≤ 16097, except p = 257. Let p_i = 2t_i + 1,q = 2t₀ + 3 be primes where t_i;i = 0,1,…, n are odd > 1 and k_i are integers ≥2. We prove that if Z-cyclic TWh(p_i) and TWh(q + 1) exist then Z-cyclic TWh(∏ⁿ_{i = 1} p_i) and TWh(q∏ⁿ_{i = 1} p_i + 1) exist. © 1996 John Wiley & Sons, Inc.     

Longtime behavior for a nonlinear wave equation arising in elasto‐plastic flow

The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in elasto‐plastic flow u_tt?div{|?u|^m?1?u}?λΔu_t+Δ²u+g(u)=f(x). It proves that under rather mild conditions, the dynamical system associated with above‐mentioned IBVP possesses a global attractor, which is connected and has finite Hausdorff and fractal dimension in the phase spaces X₁=H(Ω) × L²(Ω) and X=(H³(Ω)∩H(Ω)) × H(Ω), respectively. Copyright © 2008 John Wiley & Sons, Ltd.     

An artificial boundary condition for an advection–diffusion equation

Consider the advection–diffusion equation: u₁ + au_x1 ? vδu = 0 in ?ⁿ × ?⁺ with initial data u⁰; the Support of u⁰ is contained in ?(x₁ < 0) and a: ?ⁿ → ? is positive. In order to approximate the full space solution by the solution of a problem in ? × ?⁺, we propose the artificial boundary condition: u₁ + au_x1 = 0 on ∑. We study this by means of a transmission problem: the error is an O(v²) for small values of the viscosity v.     

Phase transitions for random walk asymptotics on free products of groups

Suppose we are given finitely generated groups Γ₁,…,Γ_m equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic or algebraic terms as singular terms up to sufficiently large order (except for some degenerate cases). We consider transient random walks on the free product Γ₁* … *Γ_m and give a complete classification of the possible asymptotic behaviour of the corresponding n‐step return probabilities. They either inherit a law of the form ?^nδn log n from one of the free factors Γ_i or obey a ?^nδn^?3/2‐law, where ? < 1 is the corresponding spectral radius and δ is the period of the random walk. In addition, we determine the full range of the asymptotic behaviour in the case of nearest neighbour random walks on free products of the form $\mathbb{Z}^{d_1}\ast \ldots \ast \mathbb{Z}^{d_m}Suppose we are given finitely generated groups Γ₁,…,Γ_m equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic or algebraic terms as singular terms up to sufficiently large order (except for some degenerate cases). We consider transient random walks on the free product Γ₁* … *Γ_m and give a complete classification of the possible asymptotic behaviour of the corresponding n‐step return probabilities. They either inherit a law of the form ?^nδn log n from one of the free factors Γ_i or obey a ?^nδn^?3/2‐law, where ? < 1 is the corresponding spectral radius and δ is the period of the random walk. In addition, we determine the full range of the asymptotic behaviour in the case of nearest neighbour random walks on free products of the form $\mathbb{Z}^{d_1}\ast \ldots \ast \mathbb{Z}^{d_m}$. Moreover, we characterize the possible phase transitions of the non‐exponential types n log n in the case Γ₁ * Γ₂. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

The Blow-up Rate for a System of Heat Equations with Non-trivial Coupling at the Boundary

We study the blow-up rate of positive radial solutions of a system of two heat equations, (u₁)_t=Δu₁(u₂)_t=Δu₂, in the ball B(0, 1), with boundary conditions Under some natural hypothesis on the matrix P=(p_ij) that guarrantee the blow-up of the solution at time T, and some assumptions of the initial data u_0i, we find that if ∥x₀∥=1 then u_i(x₀, t) goestoinfinitylike(T−t), where the α_i<0 are the solutions of (P−Id)(α₁,α₂)^t=(−1,−1)^t. As a corollary of the blow-up rate we obtain the loclaization of the blow-up set at the boundary of the domain. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations

In this paper we study the magneto-micropolar fluid equations in ℝ³, prove the existence of the strong solution with initial data in H^s(ℝ³) for , and set up its blow-up criterion. The tool we mainly use is Littlewood–Paley decomposition, by which we obtain a Beale–Kato–Majda-type blow-up criterion for smooth solution (u, ω, b) that relies on the vorticity of velocity ∇ × u only. Copyright © 2007 John Wiley & Sons, Ltd.     

Z-cyclic triplewhist tournaments—The noncompatible case,part II

Two odd primes p₁ = 2 u₁ + 1, p₂ = 2 u₂ + 1, u₁, u₂ odd, are said to be noncompatible if b₁ ≠ b₂. Let b_i ≥ 2, i = 1, 2 and denote the set {(p₁, p₂): {p₁, p₂} are noncompatible, p_i < 200} by NC. In Part 1 of this study we established the existence of Z-cyclic triplewhist tournaments on 3p₁p₂ + 1 players for all (p₁, p₂) ϵ NC. Here we extend these results and establish Z-cyclic triplewhist tournaments on 3p₁p₂ + 1 players for all (p₁, p₂) ϵ NC and for all α₁ ≥ 1, α₂ ≥ 1. It is believed that these are the first infinite classes of such triplewhist tournaments. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 189–201, 1997     

Global attractor and decay estimates of solutions to a class of nonlinear evolution equations

In this work, we prove the existence of global attractor for the nonlinear evolution equation u_tt?Δu?Δu_t?Δu_tt + g(x, u)=f(x) in X=(H²(Ω)∩H(Ω)) × (H²(Ω)∩H(Ω)). This improves a previous result of Xie and Zhong in (J. Math. Anal. Appl. 2007; 336 :54–69.) concerning the existence of global attractor in H(Ω) × H(Ω) for a similar equation. Further, the asymptotic behavior and the decay property of global solution are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

A global existence theorem for the general coagulation–fragmentation equation with unbounded kernels

In this article an existence theorem is proved for the coagulation–fragmentation equation with unbounded kernel rates. Solutions are shown to be in the space X⁺ = {c∈L¹: ∫ (1 + x)∣c(x)∣dx < ∞} whenever the kernels satisfy certain growth properties and the non-negative initial data belong to X⁺. The proof is based on weak L¹ compactness methods applied to suitably chosen approximating equations.     

Non-linear parabolic problem associated with the penetration of a magnetic field into a substance

We study the following initial and boundary value problem: In section 1, with u₀ in L²(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L²(0, T; H ∩ H²) ∩ L^∞(0, T; H) if we assume u₀ in H and f in C¹(?,?). Numerical results are given for these two cases.     

Blow‐up rate estimates for a parabolic system with multiple nonlinearities

In this paper, we study the blow‐up behaviors for the solutions of parabolic systems u_t=Δu+δ₁e, v_t=Δv+µ₁u in ?×(0, T) with nonlinear boundary conditions Here δ_i?0, µ_j?0, p_i?0, q_j?0 and at least one of δ_iµ_jp_iq_j>0(i, j=1, 2). We prove that the solutions will blow up in finite time for suitable ‘large’ initial values. The exact blow‐up rates are also obtained. Copyright © 2009 John Wiley & Sons, Ltd.     

Explicit sparse almost‐universal graphs for ${\bf {{\cal G}(n, {k \over n})}}$

For any integer n, let be a probability distribution on the family of graphs on n vertices (where every such graph has nonzero probability associated with it). A graph Γ is ‐almost‐universal if Γ satisifies the following: If G is chosen according to the probability distribution , then G is isomorphic to a subgraph of Γ with probability 1 ‐ . For any p ∈ [0,1], let (n,p) denote the probability distribution on the family of graphs on n vertices, where two vertices u and v form an edge with probability p, and the events {u and v form an edge}; u,v ∈ V (G) are mutually independent. For k ≥ 4 and n sufficiently large we construct a ‐almost‐universal‐graph on n vertices and with O(n)polylog(n) edges, where q = ? ? for such k ≤ 6, and where q = ? ? for k ≥ 7. The number of edges is close to the lower bound of Ω( ) for the number of edges in a universal graph for the family of graphs with n vertices and maximum degree k. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

A Threshold Result for a Non-local Parabolic Equation

In this paper, we consider the Cauchy problem: (ECP) u_t−Δu+p(x)u=u(x,t)∫u²(y,t)/∣x−y∣dy; x∈ℝ³, t>0, u(x, 0)=u₀(x)⩾0 x∈ℝ³, (0.2) The stationary problem for (ECP) is the famous Choquard–Pekar problem, and it has a unique positive solution ū(x) as long as p(x) is radial, continuous in ℝ³, p(x)⩾ā>0, and lim_{∣x∣→∞}p(x)=p¯>0. In this paper, we prove that if the initial data 0⩽u₀(x)⩽(≢)ū(x), then the corresponding solution u(x, t) exists globally and it tends to the zero steady-state solution as t→∞, if u₀(x)⩾(≢)ū(x), then the solution u(x,t) blows up in finite time. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

One‐sided operators in Lp (x) spaces

Boundedness of one‐sided maximal functions, singular integrals and potentials is established in L(I) spaces, where I is an interval in R . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)