共查询到20条相似文献,搜索用时 62 毫秒
1.
Fred B Weissler 《Journal of Differential Equations》1984,55(2):204-224
The initial value problem on [?R, R] is considered: ut(t, x) = uxx(t, x) + u(t, x)γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥∞ → ∞ as t → T. For the specific initial value ? = kψ, where ψ ? 0, ψxx + ψγ = 0, ψ(±R) = 0, k is sufficiently large, it is shown that if x ≠ 0, then limt → Tu(t, x) and limt → Tux(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0. 相似文献
2.
It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f’(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method. 相似文献
3.
4.
Abraham Boyarsky 《Journal of Mathematical Analysis and Applications》1978,63(2):490-501
Let xtu(w) be the solution process of the n-dimensional stochastic differential equation dxtu = [A(t)xtu + B(t) u(t)] dt + C(t) dWt, where A(t), B(t), C(t) are matrix functions, Wt is a n-dimensional Brownian motion and u is an admissable control function. For fixed ? ? 0 and 1 ? δ ? 0, we say that x?Rn is (?, δ) attainable if there exists an admissable control u such that P{xtu?S?(x)} ? δ, where S?(x) is the closed ?-ball in Rn centered at x. The set of all (?, δ) attainable points is denoted by (t). In this paper, we derive various properties of (t) in terms of K(t), the attainable set of the deterministic control system . As well a stochastic bang-bang principle is established and three examples presented. 相似文献
5.
A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, u1 + f(u)x = 0, u?Rm, is studied. These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system ut + f(u)x = v(Dux)x as ifv → 0 if D is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation u1 + f′(u0) ux = vDuxx should be well posed in L2 uniformly in v as v → 0. Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion. 相似文献
6.
Michael E Taylor 《Journal of Mathematical Analysis and Applications》1976,53(2):291-312
This paper extends a result of Fujita [On the blowing up of solutions to the Cauchy problem for ut = Δu + u1 + a, J. Faculty Science, U. of Tokyo 13 (1966), 109–124] to show that solutions u = u(t, x) for t > 0 and x?R2 to the equation ut = Δu + u2 with u(0, x) = a(x) must grow at a rate faster than exp(∥x∥2) at some finite time t, as long as a(x) is nonnegative and not almost everywhere zero. 相似文献
7.
Nathaniel Chafee 《Journal of Differential Equations》1974,15(3):522-540
We consider a parabolic partial differential equation ut = uxx + f(u), where ? ∞ < x < + ∞ and 0 < t < + ∞. Under suitable hypotheses pertaining to f, we exhibit a class of initial data φ(x), ? ∞ < x < + ∞, for which the corresponding solutions u(x, t) approach zero as t → + ∞. This convergence is uniform with respect to x on any compact subinterval of the real axis. 相似文献
8.
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
9.
For the Cauchy problem, ut = uxx, 0 < x < 1, 0 < t ? T, u(0, t) = f(t), 0 < t ? T, ux(0, t) = g(t), 0 < t ? T, a direct numerical procedure involving the elementary solution of υt = υxx, 0 < x, 0 < t ? T, υx(0, t) = g(t), 0 < t ? T, υ(x, 0) = 0, 0 < x and a Taylor's series computed from f(t) ? υ(0, t) is studied. Continuous dependence better than any power of logarithmic is obtained. Some numerical results are presented. 相似文献
10.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1999,328(4):291-296
The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form ut - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in QT =]0,T[×Ω, Ω ⊂ RN, with an initial condition u(0) = u0, where u0 is not bounded, |H(t,x, u, ξ)⩽ β|ξ|p + f(t,x) + βeλ1|u|f, |g|p/(p-1) ∈ Lr(QT) for some r = r{N) ⩾ 1, and - div(a(t,x,u, Du)) is the usual Leray-Lions operator. 相似文献
11.
《Applied Mathematics Letters》2003,16(3):425-434
The matrix Riemann-Hilbert factorization approach is used to derive the leading-order, exponentially small asymptotics as t → ± ∞ such that x/t ∼ O(1) of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation, i∂tu + ∂x2u − 2(|u|2 − 1)u = 0, with finite density initial data u(x,0) = x→±∞exp(i(1 ∓ 1)φ/2)(1+o(1)), φ ϵ [0, 2π). 相似文献
12.
V. F. Butuzov 《Computational Mathematics and Mathematical Physics》2006,46(3):413-424
A stationary solution to the singularly perturbed parabolic equation ?u t + ε2 u xx ? f(u, x) = 0 with Neumann boundary conditions is considered. The limit of the solution as ε → 0 is a nonsmooth solution to the reduced equation f(u, x) = 0 that is composed of two intersecting roots of this equation. It is proved that the stationary solution is asymptotically stable, and its global domain of attraction is found. 相似文献
13.
Hongfei Zhang 《Applicable analysis》2013,92(1-4):107-137
The singular diffusion equation ut=(u?1ux)x:arises in many areas of application, e.g. in the central limit approximation to Carleman's model of Boltzman equation, or, in the expansion of a thermalized electron cloud in plasma physics. This paper concerns the existence and uniqueness of solution of a mixed boundary value problem of equation ut=(um=1ux)x for ?1 < m ≤0. 相似文献
14.
Richard E. Ewing 《Journal of Mathematical Analysis and Applications》1979,71(1):167-186
Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: (p(x)ux)x ? q(x)u = p(x)ut, 0 < x < 1,0 < t? T; ; ; p(0) ux(0, t) = g(t), 0 < t0 ? t ? T, is ill-posed in the sense of Hadamard. Complex variable and Dirichlet series techniques are used to establish Hölder continuous dependence of the solution upon the data under the additional assumption of a known uniform bound for ¦ u(x, t)¦ when 0 ? x ? 1 and 0 ? t ? T. Numerical results are obtained for the problem where the data ?1, ?2 and g are known only approximately. 相似文献
15.
Stephanos Venakides 《纯数学与应用数学通讯》1985,38(2):125-155
The inverse scattering method is used to determine the distribution limit as ? → 0 of the solution u(x, t, ?) of the initial value problem. Ut ? 6uux + ?2uxxx = 0, u(x, 0) = v(x), where v(x) is a positive bump which decays sufficiently fast as x x→±α. The case v(x) ? 0 has been solved by Peter D. Lax and C. David Levermore [8], [9], [10]. The computation of the distribution limit of u(x, t, ?) as ? → 0 is reduced to a quadratic maximization problem, which is then solved. 相似文献
16.
Bui An Ton 《Journal of Mathematical Analysis and Applications》1975,49(3):713-720
The solution of the initial boundary-value problem u?′ ? ?D2u? + u?Du? = f on (a, b) x(0, T), u?(a, t) = u?(b, t) = 0 and u?(x, 0) = 0 on (a, b), is shown to converge to the solution of the limiting equation as the viscosity tends to zero. Estimates on the rate of convergence are given. 相似文献
17.
The direct method is applied to the two dimensional Burgers equation with a variable coefficient (u t + uu x ? u xx ) x + s(t)u yy = 0 is transformed into the Riccati equation $H' - \tfrac{1} {2}H^2 + \left( {\tfrac{\rho } {2} - 1} \right)H = 0$ via the ansatz $u\left( {x,y,t} \right) = \tfrac{1} {{\sqrt t }}H(\rho ) + \tfrac{y} {{2\sqrt t }}\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , provided that s(t) = t ?3/2. Further, a generalized Cole-Hopf transformations $u\left( {x,y,t} \right) = \tfrac{y} {{2\sqrt t }} - \tfrac{2} {{\sqrt t }}\tfrac{{U_\rho (\rho ,r)}} {{U(\rho ,r)}}$ , $\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , r(t) = log t is derived to linearize (u t + uu x ? u xx ) x + t ?3/2 u yy to the parabolic equation $U_r = U_{\rho \rho } + \left( {\tfrac{\rho } {2} - 1} \right)U_\rho$ . 相似文献
18.
V. F. Butuzov 《Computational Mathematics and Mathematical Physics》2007,47(4):620-628
The singularly perturbed parabolic equation ?u t + ε2Δu ? f(u, x, ε) = 0, x ∈ D ? ?2, t > 0 with Robin conditions on the boundary of D is considered. The asymptotic stability as t → ∞ and the global domain of attraction are analyzed for the stationary solution whose limit as ε → 0 is a nonsmooth solution to the reduced equation f(u, x, 0) = 0 that consists of two intersecting roots of this equation. 相似文献
19.
《Journal of Computational and Applied Mathematics》1999,104(2):123-143
This paper deals with the construction of analytic-numerical solutions with a priori error bounds for systems of the type ut = Auxx, u(0,t) + ux(0,t) = 0, Bu(1,t) + Cux(1,t) = 0, 0 < x < 1, t > 0, u(x,0) = f(x). Here A, B, C are matrices for which no diagonalizable hypothesis is assumed. First an exact series solution is obtained after solving appropriate vector Sturm-Liouville-type problems. Given an admissible error ε and a bounded subdomain D, after appropriate truncation an approximate solution constructed in terms of data and approximate eigenvalues is given so that the error is less than the prefixed accuracy ε, uniformly in D. 相似文献
20.
A. V. Faminskii 《Mathematical Notes》2008,83(1-2):107-115
In the strip П = (?1, 0) × ?, we establish the existence of solutions of the Cauchy problem for the Korteweg-de Vries equation u t + u xxx + uu x = 0 with initial condition either 1) u(?1, x) = ?xθ(x), or 2) u(?1, x) = ?xθ(?x), where θ is the Heaviside function. The solutions constructed in this paper are infinitely smooth for t ∈ (?1, 0) and rapidly decreasing as x → +∞. For the case of the first initial condition, we also establish uniqueness in a certain class. Similar special solutions of the KdV equation arise in the study of the asymptotic behavior with respect to small dispersion of the solutions of certain model problems in a neighborhood of lines of weak discontinuity. 相似文献