共查询到20条相似文献,搜索用时 46 毫秒
1.
Assume that f(x,y) is a continuous function defined on unit square 0≤x≤1,0≤y≤1 and having continuous partial derivatives with respect to x, of orders up to r, where r is a positive integer. In a recent work, applying Euler-Maclaurin formula, the following expansion has been proved. 相似文献
2.
HONG Jiaxing WANG Weiye 《偏微分方程(英文版)》2009,(3):234-265
In the present paper the regularity of solutions to Dirichlet problem of degenerate elliptic Monge-Ampere equations is studied. Let Ω R^2 be smooth and convex. Suppose that u ∈ C^2(^-Ω) is a solution to the following problem: det(uij) = K(x)f(x,u, Du) in Ω with u = 0 on аΩ. Then u ∈ C^∞(f)) provided that f(x,u,p) is smooth and positive in ^-Ω × R × R^2, K〉0 in Ω and near αΩ, K = d^m ^-K, where d is the distance to αΩ, m some integer bigger than 1 and ^-K smooth and positive on ^-Ω. 相似文献
3.
Yu Jun ZHU Jin Lian ZHANG Lian Fa HE 《数学学报(英文版)》2006,22(5):1321-1328
In this paper, we consider the shadowing and the inverse shadowing properties for C^1 endomorphisms. We show that near a hyperbolic set a C^1 endomorphism has the shadowing property, and a hyperbolic endomorphism has the inverse shadowing property with respect to a class of continuous methods. Moreover, each of these shadowing properties is also "uniform" with respect to C^1 perturbation. 相似文献
4.
5.
This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut+(u^2/2)x+px=εuxx, t〉0,x∈R, -αPxx+P=f(u)+α/2ux^2-1/2u^2, t〉0,x∈R, (E) with the initial data u(0,x)=u0(x)→u±, as x→±∞ (I) Here, u_ 〈 u+ are two constants and f(u) is a sufficiently smooth function satisfying f" (u) 〉 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u_ 〈 u+, the above Riemann problem admits a unique global entropy solution u^R(x/t) u^R(x/t)={u_,(f′)^-1(x/t),u+, x≤f′(u_)t, f′(u_)t≤x≤f′(u+)t, x≥f′(u+)t. Let U(t, x) be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u0(x) - U(0,x) ∈ H^1(R) and u_ 〈 u+, the above Cauchy problem (E) and (I) admits a unique global classical solution u(t, x) which tends to the rarefaction wave u^R(x/t) as → +∞ in the maximum norm. The proof is given by an elementary energy method. 相似文献
6.
We consider the problem of whether the equation △u = p(x)f(u) on RN, N ≥ 3, has a positive solution for which lim |x|→∞(x) = ∞ where f is locally Lipschitz continuous, positive, and nondecreasing on (0,oo) and satisfies ∫1∞[F (t)]^- 1/2dt = ∞ where F(t) = ∫0^tf(s)ds. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies ∫0∞ r min|x|=r P (x) dr = ∞. Conversely, we show that a necessary condition for the solution to exist is that p satisfies ∫0∞r1+ε min |x|=rp(x)dr =∞ for all ε〉0. 相似文献
7.
In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form(P_(a,b)){D~αu(x) + f(x, u(x)) = 0, x ∈(0, 1),u(0) = u(1) = 0, D~(α-3)u(0) = a, u(1) =-b,where 3 α≤ 4, Dαis the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f(x, t) =-p(x)t~σ, with σ∈(-1, 1)and p being a nonnegative continuous function that may be singular at x = 0 or x = 1and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch¨auder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem(P_(0,0)).Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a + b 0 and f(x, t) = tφ(x, t), with φ(x, t) being a nonnegative continuous function in(0, 1)×[0, ∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green's function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem(P_(a,b)), which behaves like the unique solution of the homogeneous problem corresponding to(P_(a,b)). Some examples are given to illustrate the existence results. 相似文献
8.
Fan Lili Yin Hui Zhao Huijiang 《偏微分方程(英文版)》2008,21(2):141-172
Abstract This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R+{utt-txx+ut+f(u)x=0,t〉0,x∈R+,u(0,x)=u0(x)→u+,asx→+∞,ut(0,x)=u1(x),u(t,0)=ub.For the non-degenerate case f](u+) 〈 0, it is shown in [1] that the above initialboundary value problem admits a unique global solution u(t,x) which converges to the stationary wave φ(x) uniformly in x ∈ R+ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. Moreover, by using the space-time weighted energy method initiated by Kawashima and Matsumura [2], the convergence rates (including the algebraic convergence rate and the exponential convergence rate) of u(t, x) toward φ(x) are also obtained in [1]. We note, however, that the analysis in [1] relies heavily on the assumption that f'(ub) 〈 0. The main purpose of this paper is devoted to discussing the case of f'(ub)= 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates. 相似文献
9.
CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX 总被引:4,自引:0,他引:4 下载免费PDF全文
The authors give the first convergence proof for the Lax-Friedrichs finite differencescheme for non-convex genuinely nonlinear scalar conservation laws of the formu_t f(k(x, t), u)_x = 0,where the coefficient k(x, t) is allowed to be discontinuous along curves in the (x, t)plane. In contrast to most of the existing literature on problems with discontinuouscoefficients, here the convergence proof is not based on the singular mapping approach,but rather on the div-curl lemma (but not the Young measure) and a Lax type en-tropy estimate that is robust with respect to the regularity of k(x, t). Following [14],the authors propose a definition of entropy solution that extends the classical Kruzkovdefinition to the situation where k(x, t) is piecewise Lipschitz continuous in the (x, t)plane, and prove the stability (uniqueness) of such entropy solutions, provided that theflux function satisfies a so-called crossng condition, and that strong traces of the solu-tion exist along the curves where k(x, t) is disco 相似文献
10.
Mirjana STOJANOVIC 《数学物理学报(B辑英文版)》2013,33(6):1721-1735
We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C^∞(t∈(0,∞);H2^s+2(R^n))∩C^0(t∈[0,∞);H^s(R^n)),s∈R,to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions ∫0^2p(β)D*^βu(x,t)dβ=△xu(x,t)+f(t,u(t,x)),t≥0,x∈R^n,u(0,x)=φ(x),ut(0,x)=ψ(x),(0.1) where △xis the spatial Laplace operator,D*^β is the operator of fractional differentiation in the Caputo sense and the force term F satisfies the Assumption 1 on the regularity and growth. For the weight function we take a positive-linear combination of delta distributions concentrated at points of interval (0, 2), i.e., p(β) =m∑k=1bkδ(β-βk),0〈βk〈2,bk〉0,k=1,2,…,m.The regularity of the solution is established in the framework of the space C^∞(t∈(0,∞);C^∞(R^n))∩C^0(t∈[0,∞);C^∞(R^n))when the initial data belong to the Sobolev space H2^8(R^n),s∈R. 相似文献
11.
Let T:X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f:X → Rd is a Hlder continuous function with ∫Xfdm = 0.If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ2:=σ2 (f ) such that Sfn √ n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix σ2 . Moreover, there exists a real number A > 0 such that, for any integer n ≥ 1,Π( m*( 1√ nS f n ),N (0,σ2 ) ≤A√n, where m*(1√ n Sfn)denotes the distribution of 1√ n Sfn with respect to m, and Π is the Prokhorov metric. 相似文献
12.
Yingguang SHI 《数学年刊B辑(英文版)》2012,33(5):751-766
In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =(a,b),a function G ∈ S(w):= { f:∫I | f(x)| w(x)d x < ∞} satisfying the conditions G 2j(x) ≥ 0,x ∈(a,b),j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S(w) with G ≥ 0 satisfying sup n ∑λ0knG(xkn) k=1 n<∞ instead,where the xkn ’s are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn ’s are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given. 相似文献
13.
In this paper, we prove that the second order differential equation d^2x/dt^2+x^2n_1f(x)+p(t)=0with p(t + 1) = p(t), f(x + T) = f(x) smooth and f(x) 〉 0, possesses Lagrangian stability despite of the fact that the monotone twist condition is violated. 相似文献
14.
Wei DONG Jian Tao CHEN 《数学学报(英文版)》2006,22(3):665-670
The goal of this paper is to study the multiplicity result of positive solutions of a class of degenerate elliptic equations. On the basis of the mountain pass theorems and the sub- and supersolutions argument for p-Laplacian operators, under suitable conditions on the nonlinearity f(x, s), we show the following problem:-△pu=λu^α-a(x)u^q in Ω,u│δΩ=0 possesses at least two positive solutions for large λ, where Ω is a bounded open subset of R^N, N ≥ 2, with C^2 boundary, λ is a positive parameter, Ap is the p-Laplacian operator with p 〉 1, α, q are given constants such that p - 1 〈α 〈 q, and a(x) is a continuous positive function in Ω^-. 相似文献
15.
Biharmonic equations with asymptotically linear nonlinearities 总被引:1,自引:1,他引:0
This article considers the equation △2u = f(x, u)with boundary conditions either u|(a)Ω = (a)u/(a)n|(a)Ω = 0 or u|(a)Ω = △u|(a)Ω = 0, where f(x,t) is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in RN, N > 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f(x, t). 相似文献
16.
A new Rogosinski-type kernel function is constructed using kernel function of partial sums Sn(f; t) of generalized Fourier series on a parallel hexagon domain Ω associating with threedirection partition. We prove that an operator Wn(f; t) with the new kernel function converges uniformly to any continuous function f(t) ∈ Cn(Ω) (the space of all continuous functions with period Ω) on Ω. Moreover, the convergence order of the operator is presented for the smooth approached function. 相似文献
17.
The authors study the continuity of barrier function Be(x) with respect to the parameter. A sufficient condition which makes Be(x) be continuous with respect to c is obtained, and an example of discontinuity when the condition is not satisfied is also constructed. 相似文献
18.
Let Ω IR^N, (N ≥ 2) be a bounded smooth domain, p is Holder continuous on Ω^-,
1 〈 p^- := inf pΩ(x) ≤ p+ = supp(x) Ω〈∞,
and f:Ω^-× IR be a C^1 function with f(x,s) ≥ 0, V (x,s) ∈Ω × R^+ and sup ∈Ωf(x,s) ≤ C(1+s)^q(x), Vs∈IR^+,Vx∈Ω for some 0〈q(x) ∈C(Ω^-) satisfying 1 〈p(x) 〈q(x) ≤p^* (x) -1, Vx ∈Ω ^- and 1 〈 p^- ≤ p^+ ≤ q- ≤ q+. As usual, p* (x) = Np(x)/N-p(x) if p(x) 〈 N and p^* (x) = ∞- if p(x) if p(x) 〉 N. Consider the functional I: W0^1,p(x) (Ω) →IR defined as
I(u) def= ∫Ω1/p(x)|△|^p(x)dx-∫ΩF(x,u^+)dx,Vu∈W0^1,p(x)(Ω),
where F (x, u) = ∫0^s f (x,s) ds. Theorem 1.1 proves that if u0 ∈ C^1 (Ω^-) is a local minimum of I in the C1 (Ω^-) ∩C0 (Ω^-)) topology, then it is also a local minimum in W0^1,p(x) (Ω)) topology. This result is useful for proving multiple solutions to the associated Euler-lagrange equation (P) defined below. 相似文献
1 〈 p^- := inf pΩ(x) ≤ p+ = supp(x) Ω〈∞,
and f:Ω^-× IR be a C^1 function with f(x,s) ≥ 0, V (x,s) ∈Ω × R^+ and sup ∈Ωf(x,s) ≤ C(1+s)^q(x), Vs∈IR^+,Vx∈Ω for some 0〈q(x) ∈C(Ω^-) satisfying 1 〈p(x) 〈q(x) ≤p^* (x) -1, Vx ∈Ω ^- and 1 〈 p^- ≤ p^+ ≤ q- ≤ q+. As usual, p* (x) = Np(x)/N-p(x) if p(x) 〈 N and p^* (x) = ∞- if p(x) if p(x) 〉 N. Consider the functional I: W0^1,p(x) (Ω) →IR defined as
I(u) def= ∫Ω1/p(x)|△|^p(x)dx-∫ΩF(x,u^+)dx,Vu∈W0^1,p(x)(Ω),
where F (x, u) = ∫0^s f (x,s) ds. Theorem 1.1 proves that if u0 ∈ C^1 (Ω^-) is a local minimum of I in the C1 (Ω^-) ∩C0 (Ω^-)) topology, then it is also a local minimum in W0^1,p(x) (Ω)) topology. This result is useful for proving multiple solutions to the associated Euler-lagrange equation (P) defined below. 相似文献
19.
In the present paper, we give the explicit formula of the principal part of ∑k=0^n({k}q-[n]qx)^sxk ∏m=0^n-k-1(1-q^mx) with respect to [n]q for any integer s and q ∈ (0, 1]. And, using the expressions, we obtain saturation theorems for Bn (f , qn,x) approximating to f(x) ∈ C[O, 1], 0 〈 qn ≤ 1, qn → 1. 相似文献
20.
Nikolaos S. PAPAGEORGIOU Nikolaos YANNAKAKIS 《数学学报(英文版)》2005,21(5):977-996
This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x(t) and x(t). In this first part we prove existence and relaxation theorems. We consider the case of an usc, convex valued nonlinearity and we show that for this problem the solution set is nonempty and compact in C^1 (T, H). Also we examine the Isc, nonconvex case and again we prove the existence of solutions. In addition we establish the existence of extremal solutions and by strengthening our hypotheses, we show that the extremal solutions are dense in C^1 (T, H) to the solutions of the original convex problem (strong relaxation). An example of a nonlinear hyperbolic optimal control problem is also discussed. 相似文献