共查询到20条相似文献,搜索用时 31 毫秒
1.
A. ada 《Mathematical Methods in the Applied Sciences》1996,19(3):217-233
The mixed-Neumann problem for the non-linear wave equation □u−a(u)(∣∂tu)∣2−∣∇u∣2 = fε(z) is studied. The function fε(z) = ∑k∈Kfk(z,ε−1ϕk(z),ε), ε∈[0,1], K is finite, fk(z,θk,ε) are 2π-periodic with respect to θk. The existence of solution uε on a domain z = (t,x,y)∈[0,T]×ℝ+×ℝd, d = 1 or 2, is proved when ε is sufficiently small; T does not depend on ε. By the non-linear geometric optics method the asymptotic (with respect to ε→0) solution ũ ε is constructed. The estimation for the rest ε2rε = uε−ũε is derived and the limit rε, ε→0, is studied. 相似文献
2.
Miguel Ramos 《Journal of Mathematical Analysis and Applications》2009,352(1):246-258
We study the existence, multiplicity and shape of positive solutions of the system −ε2Δu+V(x)u=K(x)g(v), −ε2Δv+V(x)v=H(x)f(u) in RN, as ε→0. The functions f and g are power-like nonlinearities with superlinear and subcritical growth at infinity, and V, H, K are positive and locally Hölder continuous. 相似文献
3.
Taras A. Mel'nyk 《Mathematical Methods in the Applied Sciences》2008,31(9):1005-1027
We consider a boundary-value problem for the Poisson equation in a thick junction Ωε, which is the union of a domain Ω0 and a large number of ε-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂νuε + εκ(uε)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as ε → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as ε → 0) in the blow-up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H1(Ωε) is proved. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
4.
Given graphs G, H, and lists L(v) ? V(H), v ε V(G), a list homomorphism of G to H with respect to the lists L is a mapping f : V(G) → V(H) such that uv ε E(G) implies f(u)f(v) ε E(H), and f(v) ε L(v) for all v ε V(G). The list homomorphism problem for a fixed graph H asks whether or not an input graph G, together with lists L(v) ? V(H), v ε V(G), admits a list homomorphism with respect to L. In two earlier papers, we classified the complexity of the list homomorphism problem in two important special cases: When H is a reflexive graph (every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP‐complete otherwise. When H is an irreflexive graph (no vertex has a loop), the problem is polynomial time solvable if H is bipartite and H is a circular arc graph, and is NP‐complete otherwise. In this paper, we extend these classifications to arbitrary graphs H (each vertex may or may not have a loop). We introduce a new class of graphs, called bi‐arc graphs, which contains both reflexive interval graphs (and no other reflexive graphs), and bipartite graphs with circular arc complements (and no other irreflexive graphs). We show that the problem is polynomial time solvable when H is a bi‐arc graph, and is NP‐complete otherwise. In the case when H is a tree (with loops allowed), we give a simpler algorithm based on a structural characterization. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 61–80, 2003 相似文献
5.
M. K. Jain R. K. Jain R. K. Mohanty 《Numerical Methods for Partial Differential Equations》1992,8(1):21-31
We attempt to obtain a two-level implicit finite difference scheme using nine spatial grid points of O(k2 + kh2 + h4) for solving the 2D nonlinear parabolic partial differential equation v1uxx + v2uyy = f(x, y, t, u, ux, uy, u1) where v1 and v2 are positive constants, with Dirichlet boundary conditions. The method, when applied to a linear diffusion-convection problem, is shown to be unconditionally stable. Computational efficiency and the results of numerical experiments are discussed. 相似文献
6.
The interest in the use of quasimodes, or almost frequencies and almost eigenfunctions, to describe asymptotics for low‐frequency and high‐frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter ε, has been recently highlighted in many papers. In this paper we deal with the low frequencies for a Steklov‐type eigenvalue homogenization problem: we consider harmonic functions in a bounded domain of ?2, and strongly alternating boundary conditions of the Dirichlet and Steklov type on a part of the boundary. The spectral parameter appears in the boundary condition on small segments Tε of size O(ε) periodically distributed along the boundary; ε also measures the periodicity of the structure. We consider associated second‐order evolution problems on spaces of traces that depend on ε, and we provide estimates for the time t in which standing waves, constructed from quasimodes, approach their solutions uε(t) as ε→0. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
7.
We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed equations of the type (ε2(x)u′′(x))=f(x,u(x))+g(x,u(x),ε(x)u′(x)), 0<x<1, with Dirichlet and Neumann boundary conditions. Here the functions ε and g are small and, hence, regarded as singular and regular functional perturbation parameters. The main tool of the proofs is a generalization (to Banach space bundles) of an implicit function theorem of R. Magnus. 相似文献
8.
In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+up+εf(x), −Δu=μv+vq+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in RN (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ1) and ε,δ∈(0,δ0), where λ1 is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ0 is a positive number. 相似文献
9.
10.
Lorenzo Bertini Alberto De Sole Davide Gabrielli Giovanni Jona‐Lasinio Claudio Landim 《纯数学与应用数学通讯》2011,64(5):649-696
Consider the viscous Burgers equation ut + f(u)x = εuxx on the interval [0,1] with the inhomogeneous Dirichlet boundary conditions u(t,0) = ρ0, u(t,1) = ρ1. The flux f is the function f(u) = u(1 − u), ε > 0 is the viscosity, and the boundary data satisfy 0 < ρ0 < ρ1 < 1. We examine the quasi‐potential corresponding to an action functional arising from nonequilibrium statistical mechanical models associated with the above equation. We provide a static variational formula for the quasi‐potential and characterize the optimal paths for the dynamical problem. In contrast with previous cases, for small enough viscosity, the variational problem defining the quasi‐potential admits more than one minimizer. This phenomenon is interpreted as a nonequilibrium phase transition and corresponds to points where the superdifferential of the quasi‐potential is not a singleton. © 2011 Wiley Periodicals, Inc. 相似文献
11.
Evangelos A. Latos Dimitrios E. Tzanetis 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):137-151
We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ut = (un)xx + lf(u)/(ò-11 f(u)dx)2{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s
n-1
f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution.
For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}. 相似文献
12.
Rieko Shimada 《Mathematical Methods in the Applied Sciences》2007,30(3):257-289
We obtain the Lp–Lq maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in ?n (n?2). The Robin condition consists of two conditions: v ? u=0 and αu+β(T(u, p)v – 〈T(u, p)v, v〉v)=h on the boundary of the domain with α, β?0 and α+β=1, where u and p denote a velocity vector and a pressure, T(u, p) the stress tensor for the Stokes flow and v the unit outer normal to the boundary of the domain. It presents the slip condition when β=1 and non‐slip one when α=1, respectively. The slip condition is appropriate for problems that involve free boundaries. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
13.
Peter Lesky 《Mathematical Methods in the Applied Sciences》1992,15(7):453-468
We study the initial-boundary value problem for ?t2u(t,x)+A(t)u(t,x)+B(t)?tu(t,x)=f(t,x) on [0,T]×Ω(Ω??n) with a homogeneous Dirichlet boundary condition; here A(t) denotes a family of uniformly strongly elliptic operators of order 2m, B(t) denotes a family of spatial differential operators of order less than or equal to m, and u is a scalar function. We prove the existence of a unique strong solution u. Furthermore, an energy estimate for u is given. 相似文献
14.
We apply the Five Functionals Fixed Point Theorem to verify the existence of at least three positive pseudo-symmetric solutions for the discrete three point boundary value problem, ?(g(?u(t-1)))+a(t))f(u(t))=0, for t∈{a+1,…,b+1} and u(a)=0 with u(v)=u(b+2) where g(v)=|v| p-2 v, p>1, for some fixed v∈{a+1,…,b+1} and σ=(b+2+v)/2 is an integer. 相似文献
15.
In this paper, we prove the relation v(t)?u(t,x)?w(t), where u(t,x) is the solution of an impulsive parabolic equations under Neumann boundary condition ∂u(t,x)/∂ν=0, and v(t) and w(t) are solutions of two impulsive ordinary equations. We also apply these estimates to investigate the asymptotic behavior of a model in the population dynamics, and it is shown that there exists a unique solution of the model which converges to the periodic solution of an impulsive ordinary equation asymptotically. 相似文献
16.
Zhifei Zhang 《Journal of Mathematical Analysis and Applications》2010,363(2):549-558
We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g′(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain. 相似文献
17.
We study the sublinear elliptic equation, −Δ u = |u|psgn u + f(x,u) in the bounded domain Ω under the zero Dirichlet boundary condition. We suppose that 0 < p < 1 and |f(x,u)| is small enough near u = 0 and do not suppose that f(x,u) is odd on u. Then we prove that this problem has infinitely many solutions.
Supported in part by the Grant-in-Aid for Scientific Research (C) (No. 16540179), Ministry of Education, Culture, Sports,
Science and Technology. 相似文献
18.
Richard Avery 《Journal of Mathematical Analysis and Applications》2003,277(2):395-404
We apply the Five Functionals Fixed Point Theorem to verify the existence of at least three positive pseudo-symmetric solutions for the three point boundary value problem, (g(u′))′+a(t)f(u)=0, u(0)=0, and u(ν)=u(1), where g(v)=|v|p−2v, with p>1 and ν∈(0,1). 相似文献
19.
Junping Shi 《纯数学与应用数学通讯》2002,55(7):815-830
We prove that Δu + f(u) = 0 has a unique entire solution u(x, y) on ?2 which has the same sign as the function xy, where f is a balanced bistable function like f(u) = u ? u3. But we neither assume f is odd nor assume the monotonicity properties of f(u)/u. Our result generalizes a previous result by Dang, Fife, and Peletier [12]. Our approach combines bifurcation methods and recent results on the qualitative properties for elliptic equations in unbounded domains by Berestycki, Caffarelli and Nirenberg [5, 6]. © 2002 Wiley Periodicals, Inc. 相似文献
20.
Caisheng Chen 《Journal of Evolution Equations》2006,6(1):29-43
In this paper, we study the global existence, L∞ estimates and decay estimates of solutions for the quasilinear parabolic system ut = div (|∇ u|m ∇ u) + f(u, v), vt = div (|∇ v|m ∇ v) + g(u,v) with zero Dirichlet boundary condition in a bounded domain Ω ⊂ RN. In particular, we find a critical value for the existence and nonexistence of global solutions to the equation ut = div (|∇ u|m ∇ u) + λ |u|α - 1 u. 相似文献