Hamiltonian cycles in 2-connected claw-free-graphs

M. Matthews and D. Sumner have proved that of G is a 2-connected claw-free graph of order n such that δ ≧ (n ? 2)/3, then G is hamiltonian. We prove that the bound for the minimum degree δ can be reduced to n/4 under the additional condition that G is not in F, where F is the set of all graphs defined as follows: any graph H in F can be decomposed into three vertex disjoint subgraphs H₁, H₂, H₃ such that , where u_i, v_i ? V(H_i), u_j v_j ? V(H_j) 1 ? i ≦ j ≦ 3. Examples are given to show that the bound n/4 is sharp. © 1995 John Wiley & Sons, Inc.     

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.     

Existence of multiple positive solutions for −Δu−μ(u/∣x∣2)=u2*−1+σf(x)

In this paper, we consider the semilinear elliptic problem where Ω??^N (N?3) is a bounded smooth domain such that 0∈Ω, σ>0 is a real parameter, and f(x) is some given function in L^∞(Ω) such that f(x)?0, f(x)?0 in Ω. Some existence results of multiple solutions have been obtained by implicit function theorem, monotone iteration method and Mountain Pass Lemma. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical solution of a time-like Cauchy problem for the wave equation

Let D ? ?ⁿ be a bounded domain with piecewise-smooth boundary, and q(x,t) a smooth function on D × [0, T]. Consider the time-like Cauchy problem Given g, h for which the equation has a solution, we show how to approximate u(x,t) by solving a well posed fourth-order elliptic partial differential equation (PDE). We use the method of quasi-reversibility to construct the approximating PDE. We derive error estimates and present numerical results.     

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.     

Incompressible navier-stokes and euler limits of the boltzmann equation

We consider solutions of the Boltzmann equation, in a d-dimensional torus, d = 2, 3, For macroscopic times τ = t/?^N, ? « 1, t ≧ 0, when the space variations are on a macroscopic scale x = ?^N?1r, N ≧ 2, x in the unit torus. Let u(x, t) be, for t ≦ t₀, a smooth solution of the incompressible Navier Stokes equations (INS) for N = 2 and of the Incompressible Euler equation (IE) for N > 2. We prove that (*) has solutions for t ≦ t₀ which are close, to O(?²) in a suitable norm, to the local Maxwellian [p/(2πT)^d/2]exp{?[v ? ?u(x,t)]²/2T } with constant density p and temperature T . This is a particular case, defined by the choice of initial values of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For N ≧ 3 these equations correspond to variable density IE while for N = 2 they involve higher-order derivatives of the density.     

On the exponentially slow motion of a viscous shock

Using formal asymptotic methods, we study the internal layer behavior associated with the following viscous shock problem in the limit ε → 0: The convex nonlinearity f(u) satisfies f(α) = f(–α). For the steady problem, we show that the method of matched asymptotic expansions fails to uniquely determine the location of the equilibrium shock layer solution. This indeterminacy, resulting from neglecting certain exponentially small effects, is eliminated by using the projection method, which exploits certain properties of the spectrum associated with the linearized operator. For the time dependent problem, we show that the viscous shock, which is formed from initial data, drifts towards the equilibrium solution on an exponentially long time interval of the order O(e^C/ε), for some C > 0. This exponentially slow behavior is analyzed by deriving an equation of motion for the location of the viscous shock. For Burgers equation (f(u) = u²/2), the results give an analytical characterization of the slow shock layer motion observed numerically in Kreiss and Kreiss; see [11]. We also show that the shock layer behavior is very sensitive to small changes in the boundary operator. In addition, using a WKB-type method, the slow viscous shock motion is studied numerically for small ε, the results comparing favorably with corresponding analytical results. Finally, we relate the slow viscous shock motion to similar slow internal layer motion for the Allen-Cahn equation.     

Explicit formula for scalar non-linear conservation laws with boundary condition

We prove an uniqueness and existence theorem for the entropy weak solution of non-linear hyperbolic conservation laws of the form , with initial data and boundary condition. The scalar function u = u(x, t), x > 0, t > 0, is the unknown; the function f = f(u) is assumed to be strictly convex. We also study the weighted Burgers' equation: α ? ? . We give an explicit formula, which generalizes a result of Lax. In particular, a free boundary problem for the flux f(u(.,.)) at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to Keyfitz.     

Bifurcation results for a class of periodic quasi-linear parabolic equations

We consider the problem where a and f are 1-periodic in t, a is positive, f satisfies appropriate decreasing conditions; smoothness of a, f, ?Ω is also assumed. Denote by λ₀ the principal eigenvalue of Δ with zero Dirichlet boundary conditions, and define . We prove: (a) if ε ≤ 0, then no non-negative periodic solution exists but zero, and any solution with continuous non-negative initial datum converges to zero uniformly as t → ∞; (b) if ε > 0, then a unique non trivial non-negative 1-periodic solution u* exists, and any solution with continuous, non-negative not identically zero initial datum approaches uniformly u* as t → ∞.     

Generalized result with a unified proof of global existence to a class of reaction‐diffusion systems

We prove in this paper a generalized result with a unified proof of global existence in time of classical solutions to a class of a reaction diffusion system with triangular diffusion matrix on a bounded domain in . The system in question is u_t=aΔu ? f(x,t,u,v), v_t=cΔu + dΔv + ρf(x,t,u,v), , t > 0 with f(x,t,0,η) = 0  and  f(x,t,ξ,η)≤Kφ(ξ)e^ση, for all  x∈Ω, t > 0, ξ≥0, η≥0; where  ρ, K  and  σ  are real positive constants. Copyright © 2016 John Wiley & Sons, Ltd.     

On the convergence rate of vanishing viscosity approximations

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t, ·) ? u^?(t, ·)‖ = O(1)(1 + t) · |ln ?| on the distance between an exact BV solution u and a viscous approximation u^?, letting the viscosity coefficient ? → 0. In the proof, starting from u we construct an approximation of the viscous solution u^? by taking a mollification u * and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed ?. Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. © 2004 Wiley Periodicals, Inc.     

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.     

Global bounded weak solutions to a degenerate quasilinear chemotaxis system with rotation

This paper deals with the quasilinear Keller–Segel system with rotation where is a bounded domain with smooth boundary, D(u) is supposed to be sufficiently smooth and satisfies D(u)≥D₀u^{m ? 1}(m≥1) and D(u)≤D₁(u + 1)^{K ? m}u^{m ? 1}(K≥1) for all u≥0 with some positive constants D₀ and D₁, and f(u) is assumed to be smooth enough and non‐negative for all u≥0 and f(0) = 0, while S(u,v,x) = (s_ij)_{n × n} is a matrix with and with l≥2, where is nondecreasing on [0,∞). It is proved that when , the system possesses at least one global and bounded weak solution for any sufficiently smooth non‐negative initial data. Copyright © 2015 John Wiley & Sons, Ltd.     

On the Nonoscillatory Behavior of Solutions of a Second Order Linear Differential Equation

In this paper, wo improve the Sturm comparison theorem and two nonoscillation criteria of Leighton and Wintner, and establish two variants of a Wintner' s nonoscillatory criterion of the second order linear differential equation where r, c : t₀,∞) →, R > 0 a. e. on t₀,∞) and 1/r, c ε L^l(t₀,b) for each b ∞ (t₀,>) for some t₀ > 0. Using these two criteria, we improve some nonoscillation criteria of Hartman. Hille. Moore. Potter. WintnEr, and Willett. These proofs are more elegant and concise than those of theirs.     

Long‐time behavior of a nonlinear viscoelastic beam equation with past history

This paper is concerned with the existence of a global attractor for the nonlinear viscoelastic beam equation with past history memory where g(u_t) is a damping like | u_t | ^ru_t and f(u) is a source term like | u | ^αu ? | u | ^βu, by considering 0 ≤ β < α and r > 0. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence results for non-autonomous multiple-fragmentation models

We investigate an initial-value problem modelling fragmentation processes where particles split into two or more pieces at a rate, γ, that not only depends on the sizes of the particles involved but also on time. The existence of non-negative, mass-conserving solutions is established by considering a truncated version of an associated non-autonomous abstract Cauchy problem. The latter has solutions of the form u(t)=U_n(t,t₀)f, t⩾t₀, where f is the known data at some fixed time t₀⩾0 and {U_n(t,s)} is a uniformly continuous evolution system. A limit evolution system {U(t,s)} is shown to exist. Depending on the form of the known data f at time t₀, the scalar-valued function u, obtained from the limit evolution system via u(x, t)=[U(t, t₀)f](x) for a.e. x>0, t⩾t₀, is a solution of either the original initial-value problem or an integral version of this problem. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

A uniqueness condition for the polyharmonic equation in free space

Consider the polyharmonic wave equation ?u + (? Δ)^mu = f in ?ⁿ × (0, ∞) with time-independent right-hand side. We study the asymptotic behaviour of u ( x , t) as t → ∞ and show that u( x , t) either converges or increases with order t^α or In t as t → ∞. In the first case we study the limit $ u_0 \left({\bf x} \right) \colone \mathop {\lim }\limits_{t \to \infty } \,u\left({{\bf x},t} \right) $ and give a uniqueness condition that characterizes u₀ among the solutions of the polyharmonic equation ( ? Δ)^mu = f in ?ⁿ. Furthermore we prove in the case 2m ? n that the polyharmonic equation has a solution satisfying the uniqueness condition if and only if f is orthogonal to certain solutions of the homogeneous polyharmonic equation.     

Gevrey Asymptotic Representation of the Solutions of Equations with One Turning Point

An ordinary differential equation of the type with parameterξ ? IRⁿ and smooth coefficients a_j,a ? C^∞([-T,T]) is studied. It is assumed that all the characteristic roots of the equation vanish at t = 0 while for t ≠ 0 they are real and distinct. The constructions of real-valued phase functions ?pH_kl (k,l = 1., m) and of amplitude functions A_jkl such that for a given s ? [-T, T] every solution u(t, ξ) of the equation can be represented as where Ψ_j(s, ξ)= D^j_tu(s,ξ), j = 0,m-1 are given.     

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.     

Asymptotic characterization of standing waves and of the static limit for a class of wave equations of higher order with a variable coefficient and time-independent incitation

We consider the equation (?1)^m?^m (p?^mu) + ?u = ? in ?ⁿ × (0, ∞) for arbitrary positive integers m and n and under the assumptions p ? 1, ? ? C(?ⁿ) and p > 0. Even if the differential operator (?1)^m?^m (p?^mu) has no eigenvalues, the solution u(x,t) may increase as t → ∞ for 2m ≥ n. For this case, we derive necessary and sufficient conditions for the convergence of u(x,t) as t → ∞. Furthermore, we characterize the functions occurring in these conditions as solutions of the homogeneous static equation (?1)^m?^m (p?^mu) = 0, which satisfy appropriate asymptotic conditions at infinity. We also give an asymptotic characterization of the static limit.