共查询到20条相似文献,搜索用时 140 毫秒
1.
Jeng-Eng Lin 《Journal of Functional Analysis》1979,31(3):321-332
Consider a smooth solution of and is C1, and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, , and for simplicity, qr ? 0. Then the local energy decays faster than exponentially. 相似文献
2.
It is well known that every weak solution (with boundary values 0) of a semilinear equation is a regular solution if ? fulfils the growth condition . Here 2m is the order of A. In this paper we weaken this condition to . This requires a technique completely different from that which may be applied in case (1). 相似文献
3.
Robert S Strichartz 《Journal of Functional Analysis》1973,12(4):341-383
The regular representation of O(n, N) acting on is decomposed into a direct integral of irreducible representations. The homogeneous space is realized as the Hyperboloid . The problem is essentially equivalent to finding the spectral resolution of a certain self-adjoint invariant differential operator □h on H, which is the tangential part of the operator □ = Δx ? Δt on Rn + N. The spectrum of □h contains a discrete part (except when N = 1) with eigenfunctions generated by restricting to H solutions of □u = 0 which vanish in the region , and a continuous part ?. As a representation of O(n, N), ? ⊕ ? is unitarily equivalent to the regular representation on L2 of the cone , and the intertwining operator is obtained by solving the equation □u = 0 with given boundary values on the cone. Explicit formulas are given for the spectral decomposition. The special case n = N = 2 gives the Plancherel formula for SL(2, R). 相似文献
4.
M van den Berg 《Journal of Functional Analysis》1984,58(2):150-156
An inequality for trace (etΔD) is proven, where ?ΔD is the Dirichlet Laplacian for horn-shaped regions D in Rn. The results of Rozenbljum and Simon for the leading asymptotics for the growth of the number of eigenvalues of the two-dimensional Dirichlet Laplacian in the regions {} are easily recovered. An example of a horn-shaped region in R2 where that asymptotics is exponential is given. 相似文献
5.
This is the second of a series of papers devoted to the study of a class of non linear Schrödinger equations of the form in n where m is a real constant and f a complex valued non linear function. Here we study the scattering theory for the pair of equations that consists of the previous one and of the equation . Under suitable assumptions of f we prove the existence of the wave operators and asymptotic completeness for a class of repulsive interactions. The assumptions of f that ensure asymptotic completeness cover the case of a single power . 相似文献
6.
Juan C. Peral 《Journal of Functional Analysis》1980,36(1):114-145
Let u(x, t) be the solution of utt ? Δxu = 0 with initial conditions . Consider the linear operator . (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in Lp if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for . (b) If n = 2k ? 1, the result is valid for . This result are sharp in the sense that for p such that we prove the existence of in such a way that . Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers and finally we get that the convolution against the kernel is bounded in H1. 相似文献
7.
Hartmut Pecher 《Journal of Functional Analysis》1985,63(1):101-122
The scattering operator which belongs to a pair of PDEs consisting of the Klein-Gordon equation and a perturbation of it by a power-like nonlinearity z.hfl;(u) is studied. It is shown that this operator can be defined on a whole neighbourhood of the origin in energy space if , where and the space dimension n ? 2 is arbitrary. 相似文献
8.
The asymptotic behaviour as t tends to +∞ of the solution of in N × +, p > 1, was studied. It was proved that the behaviour depends strongly on the sign of and also on the rate of decay of the admissible initial data u(0, x) as tends to +∞. 相似文献
9.
Tomas Schonbek 《Journal of Differential Equations》1985,56(2):290-296
New and more elementary proofs are given of two results due to W. Littman: (1) Let . The estimate cannot hold for all u?C0∞(Q), Q a cube in , some constant C. (2) Let n ? 2, p ≠ 2. The estimate cannot hold for all C∞ solutions of the wave equation □u = 0 in ; all t ?; some function C: → . 相似文献
10.
11.
Abbas Bahri 《Journal of Functional Analysis》1981,41(3):397-427
We study in this paper problems of the type , Ω bounded ? N, , (I) where is given and where . Our main result is that (I) has an infinite number of solutions for a residual set of ? in . In particular, for many n ∈ there exists an open and dense subset of ? in such that (I) has n distinct solutions for such an ? This result is to be related to the conjecture developed in [1] of the existence of an infinite number of solutions to (I). The proof relies on a general characterization of level sets for a certain class of functionals, when there are no critical value in a large enough interval. In addition to the study of problem (I), we apply this characterization to give another proof (using, e.g., Brouwer's fixed point theorem) for some classical results about even functionals and saddle points. 相似文献
12.
A.G Ramm 《Journal of Mathematical Analysis and Applications》1984,98(1):92-98
Consider the exterior boundary value problem (▽2 + K2) u = 0, in Ω, k >0. , where Γ is a smooth closed connected surface in 3, , ∝ is called the radiation pattern. We prove that when h runs through any dense set in L2(Γ) the corresponding radiation pattern ∝(k,n) runs through a dense set in L2(S2) for any k >0, where S2 is the unit sphere in 3. 相似文献
13.
Teruo Ikebe 《Journal of Functional Analysis》1975,20(2):158-177
A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R3, is obtained, where V(x) is a long-range potential: , grad , being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator from PL2(R3) onto being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ), . 相似文献
14.
J.S. Hwang 《Journal of Mathematical Analysis and Applications》1983,91(2):434-443
For any fixed 0 < π ? 2π, let D(π) be the family of all holomorphic functions in the unit disk Δ which satisfy (i)f(0) = 0 and (ii) , for all π lying on some arc Af ? ?Δ with arclength . We show that for each 0 < ε < 1, there is a π0 > 0 such that for any f?D(π) with π < π0, the Bloch and Doob norm respectively satisfy These two estimates do not hold with ε = 0. 相似文献
15.
Mario Martelli 《Journal of Mathematical Analysis and Applications》1979,69(2):496-504
The existence of a 1-periodic solution of the generalized Liénard equation x″ + g(x)x′ + f(t, x) = e(t), where g(x) is continuous, e(t) is continuous, periodic of period 1 and with mean value 0 and f is continuous, periodic of period 1 in t, is proved under one of the following conditions: (i) there exists M ? 0 such that and (ii) there exists M ? 0 such that . Earlier results of A. C. Lazer, J. Mawhin and R. Reissig are obtained as particular cases. 相似文献
16.
V.B Headley 《Journal of Mathematical Analysis and Applications》1985,108(1):283-292
Let D(?) be the Doob's class containing all functions f(z) analytic in the unit disk Δ such that f(0) = 0 and lim on an arc A of ?Δ with length . It is first proved that if f?D(?) then the spherical norm ∥ f ∥ = supz?Δ, where C1 = limn→∞. Next, U represents the Seidel's class containing all non-constant functions f(z) bounded analytic in Δ such that almost everywhere. It is proved that inff?U∥f∥ = 0, and if f has either no singularities or only isolated singularities on ?Δ, then ∥f∥ ? C1. Finally, it is proved that if f is a function normal in Δ, namely, the norm ∥f∥< ∞, then we have the sharp estimate ∥fp∥ ? p∥f∥, for any positive integer p. 相似文献
17.
David W. Bange 《Journal of Differential Equations》1975,17(1):61-72
This paper treats the quasilinear, parabolic boundary value problem u(0, t) = ?1(t); u(l, t) = ?2(t) on an infinite strip with the functions being periodic in t. The major theorem of the paper gives sufficient conditions on for this problem to have a periodic solution u(x, t) which may be constructed by successive approximations with an integral operator. Some corollaries to this theorem offer more explicit conditions on and indicate a method for determining the initial estimate at which the iteration may begin. 相似文献
18.
Six different formulations equivalent to the statement that, for n ? 2, the sum ∑k = 1n (?1)kS(n, k) ≠ 0, where the S(n, k) are Stirling numbers of the second kind, are shown to hold. Using number-theoretic methods, a sufficient condition for the above statement to be true for a set of positive integers n having density 1 is then obtained. It remains open whether it is true for all n > 2. The equivalent statements then yield information on the irreducibility of the polynomials ∑k = 1nS(n, k)tk = 1 over the rationals, the nonreal zeros for successive derivatives , a gap theorem for the nonzero coefficients of exp(?ez), and the continuous solution of the differential-difference equation , where ∥ denotes the greatest integer function. 相似文献
19.
20.
Steven Zelditch 《Journal of Functional Analysis》1983,50(1):67-80
We prove a Szegö-type theorem for some Schrödinger operators of the form with V smooth, positive and growing like . Namely, let πλ be the orthogonal projection of L2 onto the space of the eigenfunctions of H with eigenvalue ?λ; let A be a 0th order self-adjoint pseudo-differential operator relative to Beals-Fefferman weights and with total symbol a(x, ξ); and let f∈C(). Then (assuming one limit exists). 相似文献