共查询到20条相似文献,搜索用时 319 毫秒
1.
T. Meškauskas 《Lithuanian Mathematical Journal》1998,38(3):250-261
This paper deals with an initial boundary-value problem for the generalized derivative nonlinear Schrödinger equation. The cases of zero Dirichlet and generalized periodic boundary conditions are considered. The global existence of a solution inL ∞(0,∞;H b 1) is proved. The uniqueness inL ∞(0,T;H b 1)∩{u: ?u/?x εL ∞(Ω×(0,T))} is also established. 相似文献
2.
Bertrand Lemaire 《Discrete Mathematics》2006,306(15):1669-1683
We give a characterization of the non-empty binary relations ? on a N*-set A such that there exist two morphisms of N*-sets u1,u2:A→R+ verifying u1?u2 and x?y⇔u1(x)>u2(y). They are called homothetic interval orders. If ? is a homothetic interval order, we also give a representation of ? in terms of one morphism of N*-sets u:A→R+ and a map such that x?y⇔σ(x,y)u(x)>u(y). The pairs (u1,u2) and (u,σ) are “uniquely” determined by ?, which allows us to recover one from each other. We prove that ? is a semiorder (resp. a weak order) if and only if σ is a constant map (resp. σ=1). If moreover A is endowed with a structure of commutative semigroup, we give a characterization of the homothetic interval orders ? represented by a pair (u,σ) so that u is a morphism of semigroups. 相似文献
3.
José L. López Ester Pérez Sinusía 《Journal of Mathematical Analysis and Applications》2007,328(2):931-945
We consider three singularly perturbed convection-diffusion problems defined in three-dimensional domains: (i) a parabolic problem −?(uxx+uyy)+ut+v1ux+v2uy=0 in an octant, (ii) an elliptic problem −?(uxx+uyy+uzz)+v1ux+v2uy+v3uz=0 in an octant and (iii) the same elliptic problem in a half-space. We consider for all of these problems discontinuous boundary conditions at certain regions of the boundaries of the domains. For each problem, an asymptotic approximation of the solution is obtained from an integral representation when the singular parameter ?→0+. The solution is approximated by a product of two error functions, and this approximation characterizes the effect of the discontinuities on the small ?− behaviour of the solution and its derivatives in the boundary layers or the internal layers. 相似文献
4.
L.W. White 《Journal of Mathematical Analysis and Applications》1980,73(1):267-277
Let G be a bounded subset of Rn with a smooth boundary and Q = G × (0, T]. We consider a control problem governed by the Sobolev initial-value problem Myt(u) + Ly(u) = u in L2(Q), y(·, 0; u) = 0 in L2(G), where M = M(x) and L = L(x) are symmetric uniformly strongly elliptic operators of orders 2m and 2l, respectively. The problem is to find the control u0 of L2(Q)-norm at most b that steers to within a prescribed tolerance ? of a given function Z in L2(G) and that minimizes a certain energy functional. Our main results establish regularity properties of u0. We also give results concerning the existence and uniqueness of the optimal control, the controllability of Sobolev initial-value problems, and properties of the Lagrange multipliers associated with the problem constraints. 相似文献
5.
Oscar Rojo 《Journal of Mathematical Analysis and Applications》1977,61(1):208-215
This paper considers a problem proposed by Bellman in 1970: given a continuous kernel K(x, y) defined on I × I, find a pair of continuous functions f and g such that f(x) + g(y) ? K(x, y) on I × I and ∝I (f + g) is minimum. The notion of basic decomposition of K is defined, and it is shown that whenever K(x, y) or K(x, a + b ? y), I = [a, b], admits a basic decomposition, Bellman's problem has a unique differentiable solution, provided K is differentiable. Explicit formulas for such solutions are given. More generally, there are kernels which admit basic decompositions on subintervals which can be “pasted together” to define a unique piecewise differentiable solution. 相似文献
6.
R.E White 《Journal of Mathematical Analysis and Applications》1979,68(1):157-170
We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)?1?Lq((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of Lp(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x0)) u(x0), (Fu)(x) = ∫xLf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L2(0, L) function. 相似文献
7.
8.
Zhaoyun Zhang Bo‐Qing Dong Yan Jia 《Mathematical Methods in the Applied Sciences》2019,42(9):3388-3399
This paper focuses on a system of the two‐dimensional (2D) magnetohydrodynamic (MHD) equations with the partial kinematic dissipation (?yyu1,?xxu2) and the partial magnetic diffusion (?yyb1,?xxb2). Based on the basic energy estimates only, we are able to show that this system always possesses a unique global smooth solution when the initial data are sufficiently smooth. Moreover, we obtain optimal large‐time decay rates of both solutions and their higher order derivatives by developing the classic Fourier splitting methods together with the auxiliary decay estimates of the first derivative of solutions and induction technique. 相似文献
9.
《Journal of Computational and Applied Mathematics》1999,102(2):315-331
In this paper, on the basis of the results of Ishihara et al. (1997), we first discuss global convergence theorems for the improved SOR-Newton and block SOR-Newton methods with orderings applied to a system of mildly nonlinear equations, which includes as a special case the discretized version of the Dirichlet problem, for the equation ϵΔu + p(x)ux + q(y)uy = f(x, y, u), where f is continuously differentiable and fu(x, y, u) ⩾ 0. Moreover, we propose a practical choice of the multiple relaxation parameters {ωi} for them. Numerical examples are also given. 相似文献
10.
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
11.
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
12.
Yu. K. Sabitova 《Russian Mathematics (Iz VUZ)》2009,53(12):41-49
We consider the equation y
m
u
xx
− u
yy
− b
2
y
m
u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u
y
(x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u
x
(0, y) = 0 or u
x
(0, y) = u
x
(1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems 相似文献
13.
Let R be a commutative ring with unity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex set W*(R), where W*(R) is the set of all nonzero and nonunit elements of R, and two distinct vertices a and b are adjacent if and only if a ? Rb and b ? Ra, where Rc is the ideal generated by the element c in R. Recently, it has been proved that for every nonlocal finite ring R, Γ′(R) is a unicyclic graph if and only if R ? ?2 × ?4, ?3 × ?3, ?2 × ?2[x]/(x 2). We generalize the aforementioned result by showing that for every commutative ring R, Γ′(R) is a unicyclic graph if and only if R ? ?2 × ?4, ?3 × ?3, ?2 × ?2[x]/(x 2), ?2[x, y]/(x, y)2, ?4[x]/(2x, x 2). We prove that for every positive integer Δ, the set of all commutative nonlocal rings with maximum degree at most Δ is finite. Also, we classify all rings whose cozero-divisor graph has maximum degree 3. Among other results, it is shown that for every commutative ring R, gr(Γ′(R)) ∈ {3, 4, ∞}. 相似文献
14.
The aim of this paper is to investigate the behaviour as t→∞ of solutions to the Cauchy problem ut−△ut−v△u−(b,∇u)=∇⋅F(u),u(x,0)=u0(x), where v>0 is a fixed constant, t≥0, x∈Ω, Ω is a bounded domain in Rn. We will first establish an a priori estimate. Then, we establish the global existence, uniqueness and continuous dependence of the weak solution for the Sobolev-Galpern type equation with the Dirichlet boundary. 相似文献
15.
The nonlinear Klein-Gordon equation ?μ?μΦ + M2Φ + λ1Φ1?m + λ2Φ1?2m = 0 has the exact formal solution Φ = [u2m ?λ1um/(m ? 2)M2+λ12/(m?2)2M4?λ2/4(m ? 1)M2]1/mu?1, m ≠ 0, 1, 2, where u and v?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [aum + b(uv)m/2 + cvm]k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0. 相似文献
16.
R.A. Willoughby 《Linear algebra and its applications》1977,18(1):75-94
One aspect of the inverse M-matrix problem can be posed as follows. Given a positive n × n matrix A=(aij) which has been scaled to have unit diagonal elements and off-diagonal elements which satisfy 0 < y ? aij ? x < 1, what additional element conditions will guarantee that the inverse of A exists and is an M-matrix? That is, if A?1=B=(bij), then bii> 0 and bij ? 0 for i≠j. If n=2 or x=y no further conditions are needed, but if n ? 3 and y < x, then the following is a tight sufficient condition. Define an interpolation parameter s via x2=sy+(1?s)y2; then B is an M-matrix if s?1 ? n?2. Moreover, if all off-diagonal elements of A have the value y except for aij=ajj=x when i=n?1, n and 1 ? j ? n?2, then the condition on both necessary and sufficient for B to be an M-matrix. 相似文献
17.
This paper deals with the Cauchy problem for a higher order shallow water equation yt+auxy+buyx=0, where and k=2. The local well-posedness of solutions for the Cauchy problem in Sobolev space Hs(R) with s?7/2 is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are also acquired. Finally, the weak solution for the equation is considered. 相似文献
18.
In this paper, we study the nonlinear one-dimensional periodic wave equation with x-dependent coefficients u(x)ytt−(ux(x)yx)+g(x,t,y)=f(x,t) on (0,π)×R under the boundary conditions a1y(0,t)+b1yx(0,t)=0, a2y(π,t)+b2yx(π,t)=0 ( for i=1,2) and the periodic conditions y(x,t+T)=y(x,t), yt(x,t+T)=yt(x,t). Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. A main concept is the notion “weak solution” to be given in Section 2. For T is the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained. 相似文献
19.
20.
We show that an isolated singularity at the origin 0 of a smooth solution (u,p) of the stationary Navier-Stokes equations is removable if the velocity u satisfies u∈Ln or |u(x)|=o(|x|-1) as x→0. Here n?3 denotes the dimension. As a byproduct of the proof, we also obtain a new interior regularity theorem. 相似文献