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1.
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
2.
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
3.
Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem . Here the Aj's are constant, k × k Hermitian matrices, x = (x1,…, xn), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit , where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U0(t) and such that depending on ? and that the local energy of nonstatic solutions decays as . More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation at infinity. 相似文献
4.
The existence of a unique strong solution of the nonlinear abstract functional differential equation , (E) is established. X is a Banach space with uniformly convex dual space and, for is m-accretive and satisfies a time dependence condition suitable for applications to partial differential equations. The function F satisfies a Lipschitz condition. The novelty of the paper is that the solution u(t) of (E) is shown to be the uniform limit (as n → ∞) of the sequence un(t), where the functions un(t) are continuously differentiate solutions of approximating equations involving the Yosida approximants. Thus, a straightforward approximation scheme is now available for such equations, in parallel with the approach involving the use of nonlinear evolution operator theory. 相似文献
5.
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
6.
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
7.
Wolf von Wahl 《Journal of Functional Analysis》1978,27(1):118-135
This paper deals with classical solvability for all t of semilinear parabolic equations u′ + A(t)u = f(t, x, u, ▽u, …, ▽2m ? 1u). It is shown that the right side is allowed to grow faster than in ▽mu if a Hölder norm of u is known a priori. In the second part an example is given where an a priori estimate of a Hölder norm of u is available. Moreover, we give a new maximum principle. 相似文献
8.
According to a result of A. Ghizzetti, for any solution y(t) of the differential equation where , (0 ?i ? n ?1, either y(t) = 0 for t ? 1 or there is an integer r with 0 ? r ? n ? 1 such that exists and ≠0. Related results are obtained for difference and differential inequalities. A special case of the former has interesting applications in the study of orthogonal polynomials. 相似文献
9.
An elastic-plastic bar with simply connected cross section Q is clamped at the bottom and given a twist at the top. The stress function u, at a prescribed cross section, is then the solution of the variational inequality (0.1) is equal to the angle of the twist (after normalizing the units). Introducing the Lagrange multiplier λθ1, the unloading problem consists in solving the variational inequality (0.3) is the twisting angle for the unloaded bar; θ2 < θ1. Let (0.4) , and denote by the solutions of (0.1), (0.3), respectively, when K is replaced by . The following results are well known for the loading problem (0.1):(0.5) ; (0.6) the plastic set is connected to the boundary. In this paper we show that, in general, (0.7) ; (0.8) the plastic set is not connected to the boundary. That is, we construct domains Q for which (0.7) and (0.8) hold for a suitable choice of θ1, θ2. 相似文献
10.
11.
David Gurarie 《Journal of Mathematical Analysis and Applications》1985,108(1):223-229
For elliptic operators on Rn and certain of their singular perturbations relative compactness of B with respect to A is established. This result applies to the study of Lp-spectra of elliptic operators for different p. 相似文献
12.
Variational problems for the multiple integral , where and are studied. A new condition on g, called W1,p-quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of C. B. Morrey, it being shown in particular to be necessary for sequential weak lower semicontinuity of in and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in , p ? n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses. 相似文献
13.
14.
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
15.
This paper deals with asymptotic behavior for (weak) solutions of the equation , on + × Ω; u(t, x) = 0, on + × ?Ω. If and β is coercive, we prove that the solutions are bounded in the energy space, under weaker assumptions than those used by G. Prouse in a previous work. If in addition and ? is srongly almost-periodic, we prove for strongly monotone β that all solutions are asymptotically almost-periodic in the energy space. The assumptions made on β are much less restrictive than those made by G. Prouse: mainly, we allow β to be multivalued, and in the one-dimensional case β need not be defined everywhere. 相似文献
16.
Christer Borell 《Journal of Mathematical Analysis and Applications》1973,43(2):419-440
Let ψ be convex with respect to ?, B a convex body in Rn and f a positive concave function on B. A well-known result by Berwald states that (1) if ξ is chosen such that .The main purpose in this paper is to characterize those functions f : B → R+ such that (1) holds. 相似文献
17.
Elliptic operators , α a multi-index, with leading term positive and constant coefficient, and with lower order coefficients defined on or a quotient space are considered. It is shown that the Lp-spectrum of A is contained in a “parabolic region” Ω of the complex plane enclosing the positive real axis, uniformly in p. Outside Ω, the kernel of the resolvent of A is shown to be uniformly bounded by an L1 radial convolution kernel. Some consequences are: A can be closed in all Lp (1 ? p ? ∞), and is essentially self-adjoint in L2 if it is symmetric; A generates an analytic semigroup e?tA in the right half plane, strongly Lp and pointwise continuous at t = 0. A priori estimates relating the leading term and remainder are obtained, and summability , with φ analytic, is proved for , with convergence in Lp and on the Lebesgue set of ?. More comprehensive summability results are obtained when A has constant coefficients. 相似文献
18.
A regularity result for singular nonlinear elliptic systems in inverse-power weighted Sobolev spaces
P.D Smith 《Journal of Differential Equations》1984,53(2):125-138
The compactness method to weighted spaces is extended to prove the following theorem:Let H2,s1(B1) be the weighted Sobolev space on the unit ball in Rn with norm Let n ? 2 ? s < n. Let u? [H2,s1(B1) ∩ L∞(B1)]N be a solution of the nonlinear elliptic system , are uniformly continuous functions of their arguments and satisfy: . Then there exists an R1, 0 < R1 < 1, and an α, 0 < α < 1, along with a set such that (1) , (2) Ω does not contain the origin; Ω does not contain BR1, (3) is open, (4) u is ; u is LipαBR1. 相似文献
19.
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
20.
Philippe Delanoe 《Journal of Functional Analysis》1982,45(3):403-430
Let (Vn, g) be a C∞ compact Riemannian manifold without boundary. Given the following changes of metric: , where a is a fixed constant, we study the corresponding Monge-Ampère equations (1)±, (2)±. We first solve Eq. (2)?, under some simple assumptions on F?C∞. Then, using an appropriate change of functions that enables us to take advantage of the estimates just carried out for Eq. (2)?, we extend to Eq.(1)? all the results proved in our previous articles [5, 6] for the usual Monge-Ampère equation. Although equation (2)+ is not locally invertible, and does not even admit a solution for all , a similar change of functions leads to partial results about Eq. (1)+, via C2 and C3 estimates for Eq. (2)+. Eventually we give some comments and errata of our previous article (P. Delanoë, J. Funct. Anal.41 (1981), 341–353). 相似文献