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1.
Letu be a weak solution of the initial boundary value problem for the semilinear parabolic system of order 2m:u′(t)+Au(t)+f(t,.,u,..., ▽
m
u)=0. Letf satisfy controllable growth conditions. Thenu is smooth.
This result is proved by a kind of continuity method, where the timet is the parameter of continuity. 相似文献
2.
In this paper we consider operators acting on a subspace ℳ of the space L
2 (ℝm; ℂm) of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace
ℳ is defined as the orthogonal sum of spaces ℳs,k of specific Clifford basis functions of L
2(ℝm; ℂm).
Every Clifford endomorphism of ℳ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic
operators are characterized in terms of commutation relations and they transform a space ℳs,k into a similar space ℳs′,k′. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ℳ is known.
Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is
studied in detail. 相似文献
3.
Given a∈L
1(ℝ) and A the generator of an L
1-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫
−∞
t
a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L
1(ℝ) positive, nonincreasing and log-convex is already sufficient. 相似文献
4.
T. S. Kopaliani 《Ukrainian Mathematical Journal》2008,60(12):2006-2014
We point out that if the Hardy–Littlewood maximal operator is bounded on the space L
p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L
p
(ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L
p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L
p(t) (ℝ
n
), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ
n
, if and only if p(t) = const.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008. 相似文献
5.
For a triple {V, H, V*} of Hilbert spaces, we consider an evolution inclusion of the form u′(t)+A(t)u(t)+δϕ(t, u(t)) ∋
f(t), u(0) = u0, t ∈ (0, T ], where A(t) and ϕ(t, ·), t ∈ [0, T], are a family of nonlinear operators from V to V * and a family of convex lower semicontinuous functionals with common effective domain D(ϕ) ⊂ V. We indicate conditions on the data under which there exists a unique solution of the problem in the space H
1(0, T; V)∩W
∞1 (0, T;H) and the implicit Euler method has first-order accuracy in the energy norm. 相似文献
6.
S. Staněk 《Ukrainian Mathematical Journal》2008,60(2):277-298
We present existence principles for the nonlocal boundary-value problem (φ(u(p−1)))′=g(t,u,...,u(p−1), αk(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α
k: C
p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)n(φ(u(2n−)))′=f(t,u,...,u(2n−1)), u(2k)(0)=0, αku(2k)(T)+bku(2k=1)(T)=0, 0≤k≤n−1, is given.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008. 相似文献
7.
A. F. Izé 《Annali di Matematica Pura ed Applicata》1973,96(1):21-39
Summary It is studied the relationship between the solutions of the linear functional differential equations(1) (d/dx) D(xt)=L(xt) and its perturbed equation(2) [(d/dx) D(xt)−G(t, xt)]= =L(xt)+F(t, xt) and is proved, under certain hypotheses which will be precised bellow that, if μ is a simple characteristic root of(1), then there exist a σ > 0 and a non zero vector a such that system(2) has a solution satisfying
where δ(t)=αd{F(t, ϕμ)+μG(t, ϕμ)+F(t, X0G(t, ϕμ))}, ϕμ(θ)=c·exp (μθ), −r⩾θ⩾0 and α, d, X0 are given constants.
Entrata in Redazione il 5 gennaio 1972. 相似文献
8.
In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation
has a positive and bounded solution, where q, h, f ∈ C ([0, ∞), ℝ) such that q(t) ≥ 0, but ≢ 0, h(t) ≤ t, h(t) → ∞ as t → ∞, r ∈ C
(1) ([0, ∞), (0, ∞)), p ∈ C
(2) [0, ∞), ℝ), G ∈ C(ℝ, ℝ) and τ ∈ ℝ+. In our work r(t) ≡ 1 is admissible and neither we assume G is non-decreasing, xG(x) > 0 for x ≠ 0, nor we take G is Lipschitzian. Hence the results of this paper improve many recent results.
相似文献
9.
Michael T. Lacey 《Journal d'Analyse Mathématique》1995,67(1):199-206
LetT
1 andT
2 be commuting invertible ergodic measure preserving flows on a probability space (X, A, μ). For t = (u,ν) ∈ ℝ2, letT
t
=T
1
u
T
2
v
. LetS
1 denote the unit circle in ℝ2 and σ the rotation invariant unit measure on it. Then, forf∈Lp(X) withp>2, the averagesA
t
f(x) = ∫
s
1
f(T
ts
x)σ(ds) conver the integral off for a. e.x, ast tends to 0 or infinity. This extends a result of R. Jones [J], who treated the case of three or more dimensions. 相似文献
10.
Liguang Liu 《Frontiers of Mathematics in China》2007,2(4):599-611
Let ℐ(ℝn) be the Schwartz class on ℝn and ℐ∞(ℝn) be the collection of functions ϕ ∊ ℐ(ℝn) with additional property that
for all multiindices γ. Let (ℐ(ℝn))′ and (ℐ∞(ℝn))′ be their dual spaces, respectively. In this paper, it is proved that atomic Hardy spaces defined via (ℐ(ℝn))′ and (ℐ∞(ℝn))′ coincide with each other in some sense. As an application, we show that under the condition that the Littlewood-Paley
function of f belongs to L
p(ℝn) for some p ∊ (0,1], the condition f ∊ (ℐ∞(ℝn))′ is equivalent to that f ∊ (ℐ(ℝn))′ and f vanishes weakly at infinity. We further discuss some new classes of distributions defined via ℐ(ℝn) and ℐ∞(ℝn), also including their corresponding Hardy spaces.
相似文献