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.     

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.     

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.     

On the Lp–Lq maximal regularity for the linear thermoelastic plate equation in a bounded domain

This paper is concerned with the thermoelastic plate equations in a domain Ω: subject to the boundary condition: u|=D_νu|=θ|=0 and initial condition: (u, u_t, θ)|_t=0=(u₀, v₀, θ₀). Here, Ω is a bounded domain in ?ⁿ(n≧2). We assume that the boundary ?Ω of Ω is a C⁴ hypersurface. We obtain an L_p–L_q maximal regularity theorem. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Resonance phenomena for a class of partial differential equations of higher order in cylindrical waveguides

We consider a domain Ω in ?ⁿ of the form Ω = ?^l × Ω′ with bounded Ω′ ? ?^n?l. In Ω we study the Dirichlet initial and boundary value problem for the equation ? u + [(? ? ?… ? ?)^m + (? ? ?… ? ?)^m]u = fe^?iωt. We show that resonances can occur if 2m ≥ l. In particular, the amplitude of u may increase like t^α (α rational, 0<α<1) or like in t as t∞∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases.     

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.     

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.     

Global attractors for 2‐D wave equations with displacement‐dependent damping

In this paper the long‐time behaviour of the solutions of 2‐D wave equation with a damping coefficient depending on the displacement is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H(Ω) × L₂(Ω) and H²(Ω)∩H(Ω) × H(Ω). Copyright © 2009 John Wiley & Sons, Ltd.     

Über die Radon-Transformation kreissymmetrischer Funktionen und ihre Beziehung zur Sommerfeldschen Theorie der Hankelfunktionen

For the Radon transform of functions with circular symmetry an inversion formula is proved in a new and elementary way. The inversion formula combined with Fourier theory is applied to Sommer-feld's integral for H, yielding a representation of products which generalizes Nicholson's integral for |H| ².     

Global solution to the Cauchy problem in non-linear hyperbolic thermoelasticity

We prove the existence of global solutions for small data to the initial value problem for the non-linear hyperbolic system of partial differential equations describing a thermoelastic medium in a three-dimensional space under the assumption that the coefficients in the non-linear terms are smooth functions of their arguments and behave like 0(∣η∣) for k₀ ≥ 2 near the origin. The asymptotic behaviour of the solution as t → ∞ is also described.     

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.     

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     

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     

On the well‐posedness of the Cauchy problem for an MHD system in Besov spaces

This paper is devoted to the study of the Cauchy problem of incompressible magneto‐hydrodynamics system in the framework of Besov spaces. In the case of spatial dimension n?3, we establish the global well‐posedness of the Cauchy problem of an incompressible magneto‐hydrodynamics system for small data and the local one for large data in the Besov space ? (?ⁿ), 1?p<∞ and 1?r?∞. Meanwhile, we also prove the weak–strong uniqueness of solutions with data in ? (?ⁿ)∩L²(?ⁿ) for n/2p+2/r>1. In the case of n=2, we establish the global well‐posedness of solutions for large initial data in homogeneous Besov space ? (?²) for 2<p<∞ and 1?r<∞. Copyright © 2008 John Wiley & Sons, Ltd.     

Stability of Cahn‐Hilliard fronts

We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. © 1999 John Wiley & Sons, Inc.     

Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source

We study a quasilinear parabolic–elliptic Keller–Segel system involving a source term of logistic type u_t = ? ? (?(u) ? u) ? χ ? ? (u ? v) + g(u), ? Δv = ? v + u in Ω × (0,T), subject to nonnegative initial data and the homogeneous Neumann boundary condition in a bounded domain with smooth boundary, n ≥ 1, χ > 0, ? ≥ c₁s^p for s ≥ s₀ > 1, and g(s) ≤ as ? μs² for s > 0 with a,g(0) ≥ 0, μ > 0. There are three nonlinear mechanisms included in the chemotaxis model: the nonlinear diffusion, aggregation and logistic absorption. The interaction among the triple nonlinearities shows that together with the nonlinear diffusion, the logistic absorption will dominate the aggregation such that the unique classical solution of the system has to be global in time and bounded, regardless of the initial data, whenever , or, equivalently, , which enlarge the parameter range , or , required by globally bounded solutions of the quasilinear K‐S system without the logistic source. Copyright © 2013 John Wiley & Sons, Ltd.     

The red‐shift effect and radiation decay on black hole spacetimes

We consider solutions to the linear wave equation □_g? = 0 on a (maximally extended) Schwarzschild spacetime with parameter M > 0, evolving from sufficiently regular initial data prescribed on a complete Cauchy surface Σ, where the data are assumed only to decay suitably at spatial infinity. (In particular, the support of ? may contain the bifurcate event horizon.) It is shown that the energy flux F(??) of the solution (as measured by a strictly timelike T? that asymptotically matches the static Killing field) through arbitrary achronal subsets ?? of the black hole exterior region satisfies the bound F(??) ≤ C E(v + u), where v and u denote the infimum of the Eddington‐Finkelstein advanced and retarded time of ??, v₊ denotes max{1, v}, and u₊ denotes max{1, u}, where C is a constant depending only on the parameter M, and E depends on a suitable norm of the solution on the hypersurface t ? u + v = 1. (The bound applies in particular to subsets ?? of the event horizon or null infinity.) It is also shown that ? satisfies the pointwise decay estimate |?| ≤ C Ev in the entire exterior region, and the estimates |r?| ≤ C_R?E(1 + |u|)^?1/2 and |r^1/2?| ≤ C_R?Eu in the region {r ≥ R?} ∩ J⁺(Σ) for any R? > 2M. The estimates near the event horizon exploit an integral energy identity normalized to local observers. This estimate can be thought to quantify the celebrated red‐shift effect. The results in particular give an independent proof of the classical result |?| ≥ C E of Kay and Wald without recourse to the discrete isometries of spacetime. © 2009 Wiley Periodicals, Inc.     

Wave operator for the system of the Dirac–Klein–Gordon equations

We prove the existence of the wave operator for the system of the massive Dirac–Klein–Gordon equations in three space dimensions x∈ R ³ where the masses m, M>0. We prove that for the small final data , (?, ?)∈ H ^{2 + µ, 1} × H ^{1 + µ, 1}, with and , there exists a unique global solution for system (1) with the final state conditions Copyright © 2010 John Wiley & Sons, Ltd.