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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 aL 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(ts)[Au(s)+f(s,u(s))]ds for each f:ℝ×XX almost automorphic in t, uniformly in xX, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that aL 1(ℝ) positive, nonincreasing and log-convex is already sufficient.  相似文献   

4.
We point out that if the Hardy–Littlewood maximal operator is bounded on the space L p(t)(ℝ), 1 < ap(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 < ap(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.
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.
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, fC ([0, ∞), ℝ) such that q(t) ≥ 0, but ≢ 0, h(t) ≤ t, h(t) → ∞ as t → ∞, rC (1) ([0, ∞), (0, ∞)), pC (2) [0, ∞), ℝ), GC(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.
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, forfLp(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.
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   

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