Multiplicity results near the principal eigenvalue for boundary‐value problems with periodic nonlinearity

Let us consider the boundary‐value problem where g: ? → ? is a continuous and T ‐periodic function with zero mean value, not identically zero, (λ, a) ∈ ?² and ∈ C [0, π ] with ∫^π ₀ (x) sin x dx = 0. If λ ₁ denotes the first eigenvalue of the associated eigenvalue problem, we prove that if (λ, a) → (λ ₁, 0), then the number of solutions increases to infinity. The proof combines Liapunov–Schmidt reduction together with a careful analysis of the oscillatory behavior of the bifurcation equation. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic profile of solutions to a non‐linear dissipative evolution system with conservation

We investigate the asymptotic profile to the Cauchy problem for a non‐linear dissipative evolution system with conservational form (1) provided that the initial data are small, where constants α, ν are positive satisfying ν²<4α(1 ? α), α<1. In (J. Phys. A 2005; 38 :10955–10969), the global existence and optimal decay rates of the solution to this problem have been obtained. The aim of this paper is to apply the heat kernel to examine more precise behaviour of the solution by finding out the asymptotic profile. Precisely speaking, we show that, when time t → ∞ the solution and solution in the L^p sense, where G(t, x) denotes the heat kernel and is determined by the initial data and the solution to a reformulated problem obtained in Section 3, β is related to ?₊ and ?_? which are determined by (41) in Section 4. The numerical simulation is presented in the end. The motivation of this work thanks to Nishihara (Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity. Z. Angew Math Phys 2006; 57 : 604–614). Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

Uniqueness of weak solutions in critical space of the 3‐D time‐dependent Ginzburg‐Landau equations for superconductivity

We prove the uniqueness of weak solutions of the 3‐D time‐dependent Ginzburg‐Landau equations for super‐conductivity with initial data (ψ₀, A₀)∈ L² under the hypothesis that (ψ, A) ∈ L^s(0, T; L^r,∞) × (0, T; with Coulomb gauge for any (r, s) and satisfying + = 1, + = 1, ≥ , ≥ and 3 < r ≤ 6, 3 < ≤ ∞. Here L^r,∞ ≡ is the Lorentz space. As an application, we prove a uniqueness result with periodic boundary condition when ψ₀ ∈ , A₀ ∈ L³ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hyperbolic limit of the Jin‐Xin relaxation model

We consider the special Jin‐Xin relaxation model We assume that the initial data ( ) are sufficiently smooth and close to ( ) in L^∞ and have small total variation. Then we prove that there exists a solution ( ) with uniformly small total variation for all t ≥ 0, and this solution depends Lipschitz‐continuously in the L¹ norm with respect to time and the initial data. Letting , the solution converges to a unique limit, providing a relaxation limit solution to the quasi‐linear, nonconservative system These limit solutions generate a Lipschitz semigroup on a domain containing the functions with small total variation and close to . This is precisely the Riemann semigroup determined by the unique Riemann solver compatible with (0.1). © 2005 Wiley Periodicals, Inc.     

Vortex energy and 360° Néel walls in thin‐film micromagnetics

We study the vortex pattern in ultrathin ferromagnetic films of circular crosssection. The model is based on the following energy functional: for in‐plane magnetizations m: B² → S¹ in the unit disc . The avoidance of volume charges ? · m ≠ 0 in B² and surface charges m · ν ≠ 0 on δB² leads to the formation of a vortex in the limit ε → 0. At the level ε > 0 the vortex is regularized by the formation of a 360° Néel wall (a one‐dimensional transition layer with core of scale ε) concentrated along a radius of B². We derive the limiting energy of the vortex by matching upper and lower bounds. Our analysis on the lower bound is based on a dynamical system argument and an interpolation inequality with sharp leading‐order constant, while the upper bound uses the leading‐order energy for 360° Néel walls. © 2010 Wiley Periodicals, Inc.     

Parametrized Littlewood–Paley operators on Hardy and weak Hardy spaces

In this paper, we give the boundedness of the parametrized Littlewood–Paley function on the Hardy spaces and weak Hardy spaces. As the corollaries of the above results, we prove that is of weak type (1, 1) and of type (p, p) for 1 < p < 2, respectively. This results are substantial improvement and extension of some known results. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase transition for potentials of high‐dimensional wells

For a potential function that attains its global minimum value at two disjoint compact connected submanifolds N^± in , we discuss the asymptotics, as ? → 0, of minimizers u_? of the singular perturbed functional under suitable Dirichlet boundary data . In the expansion of E _? (u_?) with respect to , we identify the first‐order term by the area of the sharp interface between the two phases, an area‐minimizing hypersurface Γ, and the energy c of minimal connecting orbits between N⁺ and N^?, and the zeroth‐order term by the energy of minimizing harmonic maps into N^± both under the Dirichlet boundary condition on ?Ω and a very interesting partially constrained boundary condition on the sharp interface Γ. © 2012 Wiley Periodicals, Inc.     

Higher moments for kinetic equations: The Vlasov–Poisson and Fokker–Planck cases

Let us consider a solution f(x,v,t)?L¹(?^2N × [0,T]) of the kinetic equation where |v|^α+1 f_o,|v|^α ?L¹ (?2N × [0, T]) for some α< 0. We prove that f has a higher moment than what is expected. Namely, for any bounded set K_x, we have We use this result to improve the regularity of the local density ρ(x,t) = ∫?dν for the Vlasov–Poisson equation, which corresponds to g = E?, where E is the force field created by the repartition ? itself. We also apply this to the Bhatnagar-Gross-;Krook model with an external force, and we prove that the solution of the Fokker-Pianck equation with a source term in L² belongs to L²([0, T]; H^1/2(?)).     

Blow‐up analysis for a system of heat equations coupled via nonlinear boundary conditions

In this paper, we study a system of heat equations coupled via nonlinear boundary conditions (1) Here p, q>0. We prove that the solutions always blow up in finite time for non‐trivial and non‐negative initial values. We also prove that the blow‐up occurs only on S_R = ?B_R for Ω = B_R = {x ? ?ⁿ:|x|<R}and under some assumptions on the initial values. Copyright © 2007 John Wiley & Sons, Ltd.     

Young‐measure solutions to a generalized Benjamin–Bona–Mahony equation

We consider the evolution of microstructure under the dynamics of the generalized Benjamin–Bona–Mahony equation (1) with u: ?² → ?. If we model the initial microstructure by a sequence of spatially faster and faster oscillating classical initial data vⁿ, we obtain a sequence of spatially highly oscillatory classical solutions uⁿ. By considering the Young measures (YMs) ν and µ generated by the sequences vⁿ and uⁿ, respectively, as n → ∞, we derive a macroscopic evolution equation for the YM solution µ, and show exemplarily how such a measure‐valued equation can be exploited in order to obtain classical evolution equations for effective (macroscopic) quantities of the microstructure for suitable initial data vⁿ and non‐linearities f. Copyright © 2005 John Wiley & Sons, Ltd.     

A Petrov‐Galerkin method with quadrature for semi‐linear second‐order hyperbolic problems

We propose and analyze a Crank–Nicolson quadrature Petrov–Galerkin (CNQPG) ‐spline method for solving semi‐linear second‐order hyperbolic initial‐boundary value problems. We prove second‐order convergence in time and optimal order H² norm convergence in space for the CNQPG scheme that requires only linear algebraic solvers. We demonstrate numerically optimal order H^k, k = 0,1,2, norm convergence of the scheme for some test problems with smooth and nonsmooth nonlinearities. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Small degree out‐branchings

Using a suitable orientation, we give a short proof of a strengthening of a result of Czumaj and Strothmann 4 : Every 2‐edge‐connected graph G contains a spanning tree T with the property that for every vertex v. As an analogue of this result in the directed case, we prove that every 2‐arc‐strong digraph D has an out‐branching B such that . A corollary of this is that every k‐arc‐strong digraph D has an out‐branching B such that , where . We conjecture that in this case would be the right (and best possible) answer. If true, this would again imply a strengthening of a result from 4 concerning spanning trees with small degrees in k‐connected graphs when k ≥ 2. We prove that for acyclic digraphs the existence of an out‐branching satisfying prescribed bounds on the out‐degrees of each vertex can be checked in polynomial time. A corollary of this is that the existence of arc‐disjoint branchings , , where the first is an out‐branching rooted at s and the second an in‐branching rooted at t, can be checked in polynomial time for the class of acyclic digraphs © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 297–307, 2003     

Global existence of the radially symmetric solutions of the Navier–Stokes equations for the isentropic compressible fluids

We study the isentropic compressible Navier–Stokes equations with radially symmetric data in an annular domain. We first prove the global existence and regularity results on the radially symmetric weak solutions with non‐negative bounded densities. Then we prove the global existence of radially symmetric strong solutions when the initial data ρ₀, u ₀ satisfy the compatibility condition for some radially symmetric g ∈ L². The initial density ρ₀ needs not be positive. We also prove some uniqueness results on the strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Hyers–Ulam–Rassias stability of a generalized Apollonius–Jensen type additive mapping and isomorphisms between C *‐algebras

Let X, Y be Banach modules over a C *‐algebra. We prove the Hyers–Ulam–Rassias stability of the following functional equation in Banach modules over a unital C *‐algebra: It is shown that a mapping f: X → Y satisfies the above functional equation and f (0) = 0 if and only if the mapping f: X → Y is Cauchy additive. As an application, we show that every almost linear bijection h: A → B of a unital C *‐algebra A onto a unital C *‐algebra B is a C *‐algebra isomorphism when h (2^d uy) = h (2^d u) h (y) for all unitaries u ∈ A, all y ∈ A, and all d ∈ Z . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mean Ramsey–Turán numbers

A ρ‐mean coloring of a graph is a coloring of the edges such that the average number of colors incident with each vertex is at most ρ. For a graph H and for ρ ≥ 1, the mean Ramsey–Turán number RT(n, H,ρ ? mean) is the maximum number of edges a ρ‐mean colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. It is conjectured that where is the maximum number of edges a k edge‐colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. We prove the conjecture holds for . We also prove that . This result is tight for graphs H whose clique number equals their chromatic number. In particular, we get that if H is a 3‐chromatic graph having a triangle then . © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 126–134, 2006     

Die Lösung der Prae-Maxwellschen Gleichungen mit Hilfe vollständiger Lösungssysteme

This paper deals with the Neumann problem of the pre-Maxwell partial differential equations for a vector field v defined in a region G ? R ³. We approximate its uniquely determined solution (integrability conditions assumed) uniformly on G by explicitly computable particular integrals and linear combinations of vector fields with a “fundamental” sequence of points .     

Finite‐dimensional global attractor for a semi‐discrete fractional nonlinear Schrödinger equation

We consider a semi‐discrete in time Crank–Nicolson scheme to discretize a weakly damped forced nonlinear fractional Schrödinger equation u _t ?i (?Δ)^α u +i |u |²u +γ u =f for considered in the the whole space . We prove that such semi‐discrete equation provides a discrete infinite‐dimensional dynamical system in that possesses a global attractor in . We show also that if the external force is in a suitable weighted Lebesgue space, then this global attractor has a finite fractal dimension. Copyright © 2017 John Wiley & Sons, Ltd.     

A note on a third‐order multi‐point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we prove some existence results for the following third‐order multi‐point boundary value problem at resonance where f: [0, 1] × R³ → R is a continuous function, 0 < ξ₁ < ??? < ξ_m < 1, α_i ∈ R, i = 1, …, m, m ≥ 1 and 0 < η₁ < η₂ < ??? < η_n < 1, β_j ∈ R, j = 1, 2, …, n, n ≥ 2. In this paper, the dimension of the linear space Ker L (linear operator L is defined by Lx = x′^′′) is equal to 2. Since all the existence results for third‐order differential equations obtained in previous papers are for the case dim Ker L = 1, our work is new.     

Eulerian subgraphs in 3‐edge‐connected graphs and Hamiltonian line graphs

In this paper, we show that if G is a 3‐edge‐connected graph with and , then either G has an Eulerian subgraph H such that , or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. If G is a 3‐edge‐connected planar graph, then for any , G has an Eulerian subgraph H such that . As an application, we obtain a new result on Hamiltonian line graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 308–319, 2003