共查询到20条相似文献,搜索用时 15 毫秒
1.
Matthias Winter 《Mathematical Methods in the Applied Sciences》1995,18(2):147-168
We consider the equation (?1)m?m (p?mu) + ?u = ? in ?n × (0, ∞) for arbitrary positive integers m and n and under the assumptions p ? 1, ? ? C(?n) and p > 0. Even if the differential operator (?1)m?m (p?mu) has no eigenvalues, the solution u(x,t) may increase as t → ∞ for 2m ≥ n. For this case, we derive necessary and sufficient conditions for the convergence of u(x,t) as t → ∞. Furthermore, we characterize the functions occurring in these conditions as solutions of the homogeneous static equation (?1)m?m (p?mu) = 0, which satisfy appropriate asymptotic conditions at infinity. We also give an asymptotic characterization of the static limit. 相似文献
2.
We consider a domain Ω in ?n of the form Ω = ?l × Ω′ with bounded Ω′ ? ?n?l. In Ω we study the Dirichlet initial and boundary value problem for the equation ? u + [(? ? ?… ? ?)m + (? ? ?… ? ?)m]u = fe?iωt. We show that resonances can occur if 2m ≥ l. In particular, the amplitude of u may increase like tα (α rational, 0<α<1) or like in t as t∞∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases. 相似文献
3.
Matthias Winter 《Mathematical Methods in the Applied Sciences》1995,18(1):1-25
We consider the equation (?1)m?m (p?mu) + ?u = ? in ?n × [0, ∞] for arbitrary positive integers m and n and under the assumptions p ?1, ? ? C and p > 0. Under the additional assumption that the differential operator (?1)m?m (p?mu) has no eigenvalues we derive an asymptotic expansion for u(x,t) as t → including all terms up to order o(1). In particular, we show that for 2m ≥ n terms of the orders tα, log t, (log t)2 and tβ·log t as t → ∞ may occur. 相似文献
4.
Jean-Pierre Lohac 《Mathematical Methods in the Applied Sciences》1991,14(3):155-175
Consider the advection–diffusion equation: u1 + aux1 ? vδu = 0 in ?n × ?+ with initial data u0; the Support of u0 is contained in ?(x1 < 0) and a: ?n → ? is positive. In order to approximate the full space solution by the solution of a problem in ? × ?+, we propose the artificial boundary condition: u1 + aux1 = 0 on ∑. We study this by means of a transmission problem: the error is an O(v2) for small values of the viscosity v. 相似文献
5.
We introduce a new concept for weak solutions in Lq-spaces, 1 < q < ∞, of the Stokes system in an exterior domain Ω ? ?n, n ? 2. Defining the variational formulation in the homogeneous Sobolev space $ \mathop H\limits^.{_{0}}^{1,q} (\Omega )^n = \{ u \in L_{1{\rm oc}}^q (\overline \Omega )^n;\nabla u \in L^q (\Omega )^{n^2 },u\left| {_{\partial \Omega } = 0} \right.\},$ we prove existence and uniqueness of weak solutions for an arbitrary external force and a prescribed divergence g = div u. On the other hand, solutions in the sense of distributions which are defined by taking test functions only in C(Ω)n are not unique if q > n/(n?1). In this case, a hidden boundary condition related to the force exerted on the body may be imposed to single out a unique solution. 相似文献
6.
Nguyen Thanh Long Pham Ngoc Dinh Alain 《Mathematical Methods in the Applied Sciences》1993,16(4):281-295
We study the following initial and boundary value problem: In section 1, with u0 in L2(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L2(0, T; H ∩ H2) ∩ L∞(0, T; H) if we assume u0 in H and f in C1(?,?). Numerical results are given for these two cases. 相似文献
7.
We prove stability of the kink solution of the Cahn‐Hilliard equation ∂tu = ∂( ∂u − u/2 + u3/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. We prove stability of the kink solution of the Cahn‐Hilliard equation ∂tu = ∂( ∂u − u/2 + u3/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. © 1999 John Wiley & Sons, Inc. 相似文献
8.
Riccardo Sacco 《Numerical Methods for Partial Differential Equations》1994,10(6):715-738
In this article we deal with the solution in Ω ? R 2 of the quasi linear equation ?Δu = f(x, y, u(x, y)) subject to mixed boundary data and representing Gauss' law in a semiconductor device, where u and f are, respectively, the electrostatic potential and the space charge density after a suitable scaling. In the following we consider the associated variational problem of finding in a suitable subspace of H1(Ω) the minimum of the functional $ J(u)\, = \,\int {_\Omega } (\frac{1}{2}\left| {\nabla u\left| {^2 \, - \,{\cal F}(x,y,u)\,d\Omega,} \right.} \right. $, where $ {\cal F}(x,y,u)\, = \,\int f (x,y,\xi)\,d\xi, $ and we prove existence and uniqueness of a weak solution according to the technique of Convex Analysis. The numerical study is then carried on by a piecewise linear finite element approximation, which is proved to converge in the H1-norm to the exact solution of the variational problem; some numerical examples are also included. © 1994 John Wiley & Sons, Inc. 相似文献
9.
Jrg Witte 《Mathematische Nachrichten》1999,200(1):151-163
Properties of integral operators with weak singularities arc investigated. It is assumed that G ? ?n is a bounded domain. The boundary δG should be smooth concerning the Sobolev trace theorem. It will be proved that the integral operators $\int {_G \frac{{f\left(\Theta \right)}}{{x - y|^{n - 1} }}u\left(\nu \right)d\partial G_\nu }$ and $ \int {_{\partial G} \frac{{f\left(\Theta \right)}}{{|x - y|^{n - 1} }}u\left(y \right)d\partial G_y }$ maps Wpk(G) into Wpk+1(G) and Wpk?1(G) into Wpk/p(G), respectively, and are bounded. Here θ ∈ S ? ?n, where S is the unit sphere. Furthermore, f possesses bounded first order derivatives and is bounded on S. Then applications to first order systems are discussed. 相似文献
10.
Peter Lesky 《Mathematical Methods in the Applied Sciences》1991,14(7):483-508
Let Ω be a domain in ?n and let m? ?; be given. We study the initial-boundary value problem for the equation with a homogeneous Dirichlet boundary condition; here u is a scalar function, $ \bar D_x^m u: = (\partial _x^\alpha u)_{|\alpha | \le m} $ and certain restrictions are made on F guaranteeing that energy estimates are possible. We prove the existence of a value of T>0 such that a unique classical solution u exists on [0, T]×Ω. Furthermore, we show that T → ∞ if the data tend to zero. 相似文献
11.
P. H. Lesky 《Mathematical Methods in the Applied Sciences》2003,26(3):193-212
Let Ω denote an unbounded domain in ?n having the form Ω=?l×D with bounded cross‐section D??n?l, and let m∈? be fixed. This article considers solutions u to the scalar wave equation ?u(t,x) +(?Δ)mu(t,x) = f(x)e?iωt satisfying the homogeneous Dirichlet boundary condition. The asymptotic behaviour of u as t→∞ is investigated. Depending on the choice of f ,ω and Ω, two cases occur: Either u shows resonance, which means that ∣u(t,x)∣→∞ as t→∞ for almost every x ∈ Ω, or u satisfies the principle of limiting amplitude. Furthermore, the resolvent of the spatial operators and the validity of the principle of limiting absorption are studied. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
12.
Yang Zhijian 《Mathematical Methods in the Applied Sciences》2009,32(9):1082-1104
The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in elasto‐plastic flow utt?div{|?u|m?1?u}?λΔut+Δ2u+g(u)=f(x). It proves that under rather mild conditions, the dynamical system associated with above‐mentioned IBVP possesses a global attractor, which is connected and has finite Hausdorff and fractal dimension in the phase spaces X1=H(Ω) × L2(Ω) and X=(H3(Ω)∩H(Ω)) × H(Ω), respectively. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
13.
We study the maximal function Mf(x) = sup |f(x + y, t)| when Ω is a region in the (y,t) Ω upper half space R and f(x, t) is the harmonic extension to R+N+1 of a distribution in the Besov space Bαp,q(RN) or in the Triebel-Lizorkin space Fαp,q(RN). In particular, we prove that when Ω= {|y|N/ (N-αp) < t < 1} the operator M is bounded from F (RN) into Lp (RN). The admissible regions for the spaces B (RN) with p < q are more complicated. 相似文献
14.
A. G. Ramm 《Mathematical Methods in the Applied Sciences》1992,15(3):159-166
The Radon transform R(p, θ), θ∈Sn?1, p∈?1, of a compactly supported function f(x) with support in a ball Ba of radius a centred at the origin is given for all $ \theta \in \mathop {S^{n - 1} }\limits^\tilde $, where $ \mathop {S^{n - 1} }\limits^\tilde $ is an open set on Sn?1, and all p∈(? ∞, ∞), n≥2. An approximate formula is given to calculate f(x) from the given data. 相似文献
15.
We prove that the solution operators et (f, y){\cal e}_t (\phi , \psi ) for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space ([(H)\dot]1 ?Lr+1) ×L2(\dot {H}^1 \cap L_{\rho +1}) \times L_2 to [(H)\dot]sq¢\dot {H}^s_{q'} for t 1 0t\neq 0, and 0 £ s £ 1,0\leq s\leq 1, (n+1)/(1/2-1/q¢) = 1(n+1)/(1/2-1/q')= 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here et(f, y)=u(·, t){\cal e}_t(\phi , \psi )=u(\cdot , t), where u is a solution of {