共查询到20条相似文献,搜索用时 15 毫秒
1.
Xiu-Fang Ren 《中国科学 数学(英文版)》2010,53(12):3067-3084
In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic function near u = 0 with f (0) = 0, f (0) = 0, and the Floquet multiplier Mσ is defined as Mσe inx = σne inx , with σn = σ, when n 0, otherwise, σn = 0. It is proved that for each given 0 σ 1, and each given integer b 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, corresponding to b-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, these b-dimensional Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method. 相似文献
2.
In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(2|u|)u=0 on [0,π]×R under the boundary conditions a1u(t,0)−b1ux(t,0)=0, a2u(t,π)+b2ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system. 相似文献
3.
We establish the critical Fujita exponents for the solution of the porous medium equation ut=Δum, x∈R+N, t>0, subject to the nonlinear boundary condition −∂um/∂x1=up, x1=0, t>0, in multi-dimension. 相似文献
4.
Kosuke Ono 《Mathematical Methods in the Applied Sciences》2000,23(6):535-560
We study the global existence, asymptotic behaviour, and global non‐existence (blow‐up) of solutions for the damped non‐linear wave equation of Kirchhoff type in the whole space: utt+ut=(a+b∥∇u∥2γ)Δu+∣u∣αu in ℝN×ℝ+ for a, b⩾0, a+b>0, γ⩾1, and α>0, with initial data u(x, 0)=u0(x) and ut(x, 0)=u1(x). Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
5.
In this paper, we propose a new high accuracy numerical method of O(k2 + k2h2 + h4) based on off-step discretization for the solution of 3-space dimensional non-linear wave equation of the form utt = A(x,y,z,t)uxx + B(x,y,z,t)uyy + C(x,y,z,t)uzz + g(x,y,z,t,u,ux,uy,uz,ut), 0 < x,y,z < 1,t > 0 subject to given appropriate initial and Dirichlet boundary conditions, where k > 0 and h > 0 are mesh sizes in time and space directions respectively. We use only seven evaluations of the function g as compared to nine evaluations of the same function discussed in and . We describe the derivation procedure in details of the algorithm. The proposed numerical algorithm is directly applicable to wave equation in polar coordinates and we do not require any fictitious points to discretize the differential equation. The proposed method when applied to a telegraphic equation is also shown to be unconditionally stable. Comparative numerical results are provided to justify the usefulness of the proposed method. 相似文献
6.
Steven Schochet 《Journal of Differential Equations》2003,192(1):134-154
The blow-up of solutions to the PDE ψ(x)ut=[∇·A(x)∇+b(x)]um is studied via energy methods. The key step is a similarity transformation of the original unstable equation to a nonlocal stable one. 相似文献
7.
We consider a dissipative version of the modified Korteweg-de Vries equation ut+uxxx−uxx+x(u3)=0. We prove global well-posedness results on the associated Cauchy problem in the Sobolev spaces Hs(R) for s>−1/4 while for s<−1/2 we prove some ill-posedness issues. 相似文献
8.
In this paper we study the large time behavior of non-negative solutions to the Cauchy problem of ut=Δum−uq in RN×(0,∞), where m>1 and q=qc≡m+2/N is a critical exponent. For non-negative initial value u(x,0)=u0(x)∈L1(RN), we show that the solution converges, if u0(x)(1+|x|)k is bounded for some k>N, to a unique fundamental solution of ut=Δum, independent of the initial value, with additional logarithmic anomalous decay exponent in time as t→∞. 相似文献
9.
For the equation K(t)u
xx
+ u
tt
− b
2
K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t|
m
, m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability
of the boundary value problem u(0, t) = u(1, t), u
x
(0, t) = u
x
(1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1. 相似文献
10.
Joo-Paulo Dias Mrio Figueira Luis Sanchez 《Mathematical Methods in the Applied Sciences》1998,21(12):1107-1113
In this paper we consider the Cauchy problem for the equation ∂u/∂t + u ∂u/∂x + u/x = 0 for x > 0, t ⩾ 0, with u(x, 0) = u0−(x) for x < x0, u(x, 0) = u0+(x) for x > x0, u0−(x0) > u0+(x0). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u0− and u0+, a global shock front weak solution u(x, t) = u−(x, t) for x < ϕ(t), u(x, t) = u+(x, t) for x > ϕ(t), where u− and u+ are the strong solutions corresponding (respectively) to u0− and u0+ and the curve t → ϕ(t) is defined by dϕ/dt (t) = 1/2[u−(ϕ(t), t) + u+(ϕ(t), t)], t ⩾ 0 and ϕ(0) = x0. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd. 相似文献
11.
This paper deals with ut = Δu + um(x, t)epv(0,t), vt = Δv + uq(0, t)env(x,t), subject to homogeneous Dirichlet boundary conditions. The complete classification on non-simultaneous and simultaneous blow-up is obtained by four sufficient and necessary conditions. It is interesting that, in some exponent region, large initial data u0(v0) leads to the blow-up of u(v), and in some betweenness, simultaneous blow-up occurs. For all of the nonnegative exponents, we find that u(v) blows up only at a single point if m > 1(n > 0), while u(v) blows up everywhere for 0 ? m ? 1 (n = 0). Moreover, blow-up rates are considered for both non-simultaneous and simultaneous blow-up solutions. 相似文献
12.
《Journal of Differential Equations》1987,69(3):368-403
We prove the uniqueness (as well as the existence and regularity) of solutions of the Cauchy problem and of the first and mixed boundary value problems for the equation ut = φ(u)xx + b(u)x. (E) φ and b are assumed to belong to a large class of functions, including, in particular, cases φ(u) = um, b(u) = uλ, m ⩾ 1 and λ > 0. 相似文献
13.
Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations II
We construct spike layered solutions for the semilinear elliptic equation −ε2Δu+V(x)u=K(x)up−1 on a domain Ω⊂RN which may be bounded or unbounded. The solutions concentrate simultaneously on a finite number of m-dimensional spheres in Ω. These spheres accumulate as ε→0 at a prescribed sphere in Ω whose location is determined by the potential functions V,K. 相似文献
14.
Ruying Xue 《Journal of Mathematical Analysis and Applications》2006,316(1):307-327
We consider the local and global existence of solutions for a generalized Boussinesq equation utt−uxx+uxxxx+(uk+1)xx=0, k>4, with initial data in some homogenous Besov-type space. 相似文献
15.
In this paper, one-dimensional (1D) nonlinear Schrödinger equation
iut−uxx+mu+4|u|u=0 相似文献
16.
G. Gripenberg S.-O. Londen J. Prüss 《Mathematical Methods in the Applied Sciences》1997,20(16):1427-1448
It is proved that there is a (weak) solution of the equation ut=a*uxx+b*g(ux)x+f, on ℝ+ (where * denotes convolution over (−∞, t)) such that ux is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t−α, b(t)=t−β, and g(ξ)∼sign(ξ)∣ξ∣γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. 相似文献
17.
Luiz Gustavo Farah 《Journal of Differential Equations》2010,249(8):1968-1985
We prove that the initial value problem (IVP) for the critical generalized KdV equation ut+uxxx+x(u5)=0 on the real line is globally well-posed in Hs(R) if s>3/5 with the appropriate smallness assumption on the initial data. 相似文献
18.
Georg Menz 《Journal of Differential Equations》2007,242(1):171-191
First, we consider the linear wave equation utt−uxx+a(x)ut+b(x)u=0 on a bounded interval (0,L)⊂R. The damping function a is allowed to change its sign. If is positive and the spectrum of the operator (∂xx−b) is negative, exponential stability is proved for small . Explicit estimates of the decay rate ω are given in terms of and the largest eigenvalue of (∂xx−b). Second, we show the existence of a global, small, smooth solution of the corresponding nonlinear wave equation utt−σx(ux)+a(x)ut+b(x)u=0, if, additionally, the negative part of a is small enough compared with ω. 相似文献
19.
Philippe Bénilan 《Journal of Differential Equations》2004,196(2):301-315
In this paper, we study the singular limit of the Porous Medium equation ut=Δum+g(x,u), as m→∞, in a bounded domain with Neumann boundary condition. 相似文献
20.
Yuan-Wei QiMing-Xing Wang 《Journal of Mathematical Analysis and Applications》2002,267(1):264-280
In this paper we study the critical exponents of the Cauchy problem in Rn of the quasilinear singular parabolic equations: ut = div(|∇u|m − 1∇u) + ts|x|σup, with non-negative initial data. Here s ≥ 0, (n − 1)/(n + 1) < m < 1, p > 1 and σ > n(1 − m) − (1 + m + 2s). We prove that pc ≡ m + (1 + m + 2s + σ)/n > 1 is the critical exponent. That is, if 1 < p ≤ pc then every non-trivial solution blows up in finite time, but for p > pc, a small positive global solution exists. 相似文献