Neumann Initial Boundary Value Problem Research Articles

Global existence and asymptotic behavior of solutions to the homogeneous Neumann problem for ILW equation on a half-line

We consider the homogeneous Neumann initial-boundary value problem for intermediate long-wave (ILW) equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.

Removability of an isolated singularity of solutions of the Neumann problem for quasilinear parabolic equations with absorption that admit double degeneration

We consider the Neumann initial boundary-value problem for the equation $$ {u_t} = {\text{div}}\left( {{u^{m - 1}}{{\left| {Du} \right|}^{\lambda - 1}}Du} \right) - {u^p} $$ in domains with noncompact boundary and with initial Dirac delta function. In the case of slow diffusion (m + λ − 2 > 0) and critical absorption exponent (p = m + λ − 1 + (λ + 1)/N), we prove that the singularity at the point (0, 0) is removable.

Long-time behaviour of two-phase solutions to a class of forward-backward parabolic equations

We consider two-phase solutions to the Neumann initial-boundary value problem for the parabolic equation ut =[ϕ( u )] xx , where ϕ is a nonmonotone cubic-like function. First, we prove global existence for a restricted class of initial data _u_0, showing that two-phase solutions can be obtained as limiting points of the family of solutions to the Neumann initial-boundary value problem for the regularized equation _ut_ε = [ϕ(_u_ε)] xx + ε_utxx_ε (ε > 0). Then, assuming global existence, we study the long-time behaviour of two-phase solutions for any initial datum _u_0.

Long-Time Behavior of Solutions to a Class of Forward-Backward Parabolic Equations

We consider weak entropy measure-valued solutions of the Neumann initial-boundary value problem for the equation $u_t=[\phi(u)]_{xx}$, where $\phi$ is nonmonotone. These solutions are obtained from the corresponding problem for the regularized equation $u_t=[\phi(u)]_{xx}+\varepsilon u_{xxt}$ ($\varepsilon>0$) by a vanishing viscosity method and satisfy a family of suitable entropy inequalities. Relying on a strong version of these inequalities, we prove exhaustive results concerning the long-time behavior of solutions.

Random walk methods for scalar transport problems subject to Dirichlet, Neumann and mixed boundary conditions

Three stochastic-based methods are proposed for solving unsteady scalar transport problems in bounded, single-phase domains. The first (Method I), a local solution appropriate to problems having Dirichlet conditions, adapts a well-known local stochastic solution of a backward Fokker–Planck equation to scalar transport. Method II, a local solution applicable to Dirichlet, Neumann and/or mixed initial boundary value problems (IBVPs), and representing a time-dependent extension of a recently reported heuristic steady solution, provides a straightforward addition to the limited collection of techniques available for Neumann and mixed problems. This approach is shown to be equivalent to a long-standing, rigorous low-order solution and, in addition, allows development of a probabilistic-based analytical solution to Neumann problems, stated in terms of an exit probability. Method III, a global solution, likewise suitable for Neumann and mixed IBVPs, follows by combined application of domain boundary Taylor expansions and Method I. This approach is shown to be computationally equivalent to a global version of Method II.

Resonance phenomena in compound cylindrical waveguides

AbstractWe study the large time asymptotics of the solutions u(x,t) of the Dirichlet and the Neumann initial boundary value problem for the wave equation with time‐harmonic right‐hand side in domains Ω which are composed of a finite number of disjoint half‐cylinders Ω1,…,Ωr with cross‐sections Ω′1,…,Ω′r and a bounded part (‘compound cylindrical waveguides’). We show that resonances of orders t and t1/2 may occur at a finite or countable discrete set of frequencies ω, while u(x,t) is bounded as t→∞ for the remaining frequencies. A resonance of order t occurs at ω if and only if ω2 is an eigenvalue of the Laplacian −Δ in Ω with regard to the given boundary condition u=0 or ∂u/∂n=0, respectively. A resonance of order t1/2 occurs at ω if and only if (i) ω2 is an eigenvalue of at least one of the Laplacians for the cross‐sections Ω′1,…,Ωr′ with regard to the respective boundary condition and (ii) the respective homogeneous boundary value problem for the reduced wave equation ΔU+ω2U=0 in Ω has non‐trivial solutions with suitable asymptotic properties as | x | →∞ (‘standing waves’). Copyright © 2006 John Wiley & Sons, Ltd.

Neumann problem for the Korteweg–de Vries equation

We consider Neumann initial-boundary value problem for the Korteweg–de Vries equation on a half-line (0.1) { u t + λ u u x + u x x x = 0 , t > 0 , x > 0 , u ( x , 0 ) = u 0 ( x ) , x > 0 , u x ( 0 , t ) = 0 , t > 0 . We prove that if the initial data u 0 ∈ H 1 0 , 21 4 ∩ H 2 1 , 7 2 and the norm ‖ u 0 ‖ H 1 0 , 21 4 + ‖ u 0 ‖ H 2 1 , 7 2 ⩽ ε , where ε > 0 is small enough H p s , k = { f ∈ L 2 ; ‖ f ‖ H p s , k = ‖ 〈 x 〉 k 〈 i ∂ x 〉 s f ‖ L p < ∞ } , 〈 x 〉 = 1 + x 2 and λ ∫ 0 ∞ x u 0 ( x ) d x = λ θ < 0 . Then there exists a unique solution u ∈ C ( [ 0 , ∞ ) , H 2 1 , 7 2 ) ∩ L 2 ( 0 , ∞ ; H 2 2 , 3 ) of the initial-boundary value problem (0.1). Moreover there exists a constant C such that the solution has the following asymptotics u ( x , t ) = C θ ( 1 + η log t ) −1 t − 2 3 A i ′ ( x t 3 ) + O ( ε 2 t − 2 3 ( 1 + η log t ) − 6 5 ) for t → ∞ uniformly with respect to x > 0 , where η = − 9 θ λ ∫ 0 ∞ A i ′ 2 ( z ) d z and A i ( q ) is the Airy function A i ( q ) = 1 2 π i ∫ − i ∞ i ∞ e − z 3 + z q d z = 1 π Re ∫ 0 ∞ e − i ξ 3 + i ξ q d ξ .

Spatial critical points not moving along the heat flow II: The centrosymmetric case

We consider solutions of initial-boundary value problems for the heat equation on bounded domains in $\mathbb{R}^N,$ and their spatial critical points as in the previous paper [MS]. In Dirichlet, Neumann, and Robin homogeneous initial-boundary value problems on bounded domains, it is proved that if the origin is a spatial critical point never moving for sufficiently many compactly supported initial data being centrosymmetric with respect to the origin, then the domain must be centrosymmetric with respect to the origin. Furthermore, we consider spatial zero points instead of spatial critical points, and prove some similar symmetry theorems. Also, it is proved that these symmetry theorems hold for initial-boundary value problems for the wave equation.

The spatial critical points not moving along the heat flow

We consider solutions of the heat equation, in domains inRN, and their spatial critical points. In particular, we show that a solutionu has a spatial critical point not moving along the heat flow if and only ifu satisfies some balance law. Furthermore, in the case of Dirichlet, Neumann, and Robin homogeneous initial-boundary value problems on bounded domains, we prove that if the origin is a spatial critical point never moving for sufficiently many compactly supported initial data satisfying the balance law with respect to the origin, then the domain must be a ball centered at the origin.

DECAY BOUNDS FOR SOLUTIONS OF SECOND ORDER PARABOLIC PROBLEMS AND THEIR DERIVATIVES

For a class of quasilinear parabolic equations, we derive in this paper a new maximum principle for the derivatives of solutions of homogeneous Dirichlet and Neumann initial boundary value problems. In the linear heat equation this principle is used to derive new explicit decay bounds for the absolute value of the gradient of the solution, and in the quasilinear problems, criteria are determined which imply that solutions decay exponentially. Explicit decay bounds are then established.

Ultrasonic inspection devices : Dr J.U.H. Krautkramer Ges fur Elektrophysik, British Patent 1,224,821 (1971) Filed 6 March 1969

The quasilinear chemotaxis system is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, n≥2, with smooth boundary, where the focus is on cases when herein the diffusivity D(s) decays exponentially as s→∞.It is shown that under the subcriticality condition that with some C>0 and α<2n, for all suitably regular initial data satisfying an essentially explicit smallness assumption on the total mass ∫Ωu0, the corresponding Neumann initial–boundary value problem for (⋆) possesses a globally defined bounded classical solution which moreover approaches a spatially homogeneous steady state in the large time limit. Viewed as a complement of known results on the existence of small-mass blow-up solutions in cases when in (0.1) the reverse inequality holds with some α>2n, this confirms criticality of the exponent α=2n in (0.1) with regard to the singularity formation also for arbitrary n≥2, thereby generalizing a recent result on unconditional global boundedness in the two-dimensional situation.As a by-product of our analysis, without any restriction on the initial data, we obtain boundedness and stabilization of solutions to a so-called volume-filling chemotaxis system involving jump probability functions which decay at sufficiently large exponential rates.