共查询到20条相似文献,搜索用时 656 毫秒
1.
LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC
0-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can
be summarized roughly as follows:
We also show that if lim sup
t→0+t
p ‖T′(t)‖<∞ for a givenp ε [1, ∞), then lim sup
t→0+t
p‖S′(t)‖<∞; it was known previously that if limsup
t→0+t
p‖T′(t)‖<∞, then {S(t) |t ≥ 0} is differentiable and limsup
t→0+t
2p–1‖S′(t)‖<∞. 相似文献
(i) | If lim sup t→0+t log‖T′(t)‖/log(1/2) = 0 then {S(t) |t ≥ 0} is differentiable. |
(ii) | If 0<L=lim sup t→0+t log‖T′(t)‖/log(1/2)<∞ thent ↦S(t ) is differentiable on (L, ∞) in the uniform operator topology, but need not be differentiable near zero |
(iii) | For each function α: (0, 1) → (0, ∞) with α(t)/log(1/t) → ∞ ast ↓ 0, Renardy’s example can be adjusted so that limsup t→0+t log‖T′(t)‖/α(t) = 0 andt →S(t) is nowhere differentiable on (0, ∞). |
2.
LetX be a complex Banach space and letT be a bounded linear operator onX. Denote by σ
p
(T) the point spectrum ofT and by
the unit circle. We investigate how the growth of the sequence ‖T
n
‖ is influenced by the size of the set
(T) and by the geometry of the spaceX. We also prove analogous results forC
0-semigroups(T
t
)t≥0.
Research partially supported by grants from NSERC, FQRNT and the Canada research chairs program. 相似文献
3.
Ioan I. Vrabie 《Israel Journal of Mathematics》1979,32(2-3):221-235
LetX be a real Banach space,U ⊂X a given open set,A ⊂X×X am-dissipative set andF:C(0,a;U) →L
∞(0,a;X) a continuous mapping. Assume thatA generates a nonlinear semigroup of contractionsS(t): {ie221-2}) → {ie221-3}), strongly continuous at the origin, withS(t) compact for allt>0. Then, for eachu
0 ∈ {ie221-4}) ∩U there existsT ∈ ]0,a] such that the following initial value problem: (du(t))/(dt) ∈Au(t) +F(u)(t),u(0)=u
0, has at least one integral solution on [0,T]. Some extensions and applications are also included. 相似文献
4.
Osamu Hatori Takeshi Miura Rumi Shindo Hiroyuki Takagi 《Rendiconti del Circolo Matematico di Palermo》2010,59(2):161-183
Let $
A
$
A
and ℬ be unital semisimple commutative Banach algebras. It is shown that if surjections S,T: $
A
$
A
→ ℬ with S(1)=T(1)= 1 and α ∈ ℂ \ {0} satisfy r(S(a)T(b) − α)= r(ab− α) for all a,b ∈ $
A
$
A
, then S=T and S is a real algebra isomorphism, where r(a) is the spectral radius of a. Let I be a nonempty set, A and B be uniform algebras. Let ρ, τ: I → A and S,T: I → B be maps satisfying σ
π
(S(p)T(q)) ⊂ σ
π
(ρ(p) τ(q)) for all p,q ∈ I, where σ
π
(f) is the peripheral spectrum of f. Suppose that the ranges ρ(I), τ(I) ⊂ A and S(I),T(I) ⊂ B are closed under multiplication in a sense, and contain peaking functions “enough”. There exists a homeomorphism ϕ: Ch(B)→Ch(A) such that S(p)(y)= ρ(p)(ϕ(y)) and T(p)(y)= τ(p)(ϕ(y)) for every p ∈ I and y ∈ Ch(B), where Ch(A) is the Choquet boundary of A. 相似文献
5.
Christian LeMerdy 《Semigroup Forum》1998,56(2):205-224
t )t≥0 on a Hilbert space H, we establish conditions under which (Tt)t≥0 is similar to a contraction semigroup, i.e., there exists an isomorphism S Ε B (H) such that (S-1 Tt S)t≥0 is a contraction semigroup. In the case when the generator -A of (Tt)t≥0 is one-to-one, we obtain that (Tt)t≥0 is similar to a contraction semigroup if and only if A admits bounded imaginary powers. This characterizes one-to-one operators
of type strictly less than π/2 on H which belong to BIP (H). 相似文献
6.
Clustering of linearly interacting diffusions and universality of their long-time limit distribution
J. M. Swart 《Probability Theory and Related Fields》2000,118(4):574-594
Let K⊂ℝ
d
(d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K
Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX
i
(t) = ∑
j
a(j−i) (X
j
(t) −X
i
(t))dt + σ (X
i
(t))dB
i
(t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a
S
(i) = a(i) + a(−i) is recurrent, then each component X
i
(∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a
S
is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of
the components. This distribution turns out to be universal in all recurrent kernels a
S
on Abelian groups Λ.
Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000 相似文献
7.
Jerome A. Goldstein 《Annali dell'Universita di Ferrara》1971,16(1):39-43
Summary We construct nontrivial bounded solutions of an abstract evolution equationu'(t)=Au(t) (—∞<t<∞) whereA generates a (C
0) semigroup of operators {T
t;t≥0} such thatT
t converges strongly to zero ast→∞.
Supported by National Science Foundation grant GP-12722. 相似文献
Riassunto Si costruiscono soluzioni limitate non triviali di una equazioneu'(t)=Au(t) (—∞<t<∞) doveA è il generatore di un semigruppo di operatori {T t;t≥0} tale cheT t converge fortemente verso 0 pert→∞.
Supported by National Science Foundation grant GP-12722. 相似文献
8.
I. Kiguradze 《Georgian Mathematical Journal》1994,1(5):487-494
The properties of solutions of the equationu″(t) =p
1(t)u(τ1(t)) +p
2(t)u′(τ2(t)) are investigated wherep
i
:a, + ∞[→R (i=1,2) are locally summable functions τ1 :a, + ∞[→R is a measurable function, and τ2 :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ
i
(t) ≥t (i = 1,2),p
1(t)≥0,p
2
2
(t) ≤ (4 - ɛ)τ
2
′
(t)p
1(t), ɛ =const > 0 and
. In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear
space and for any such solution to vanish at infinity it is necessary and sufficient that
. 相似文献
9.
Wen heng Wang 《数学学报(英文版)》2002,18(4):727-736
Let {W(t); t≥ 0} be a standard Wiener process and S be the Strassen set of functions. We investigate the exact rates of convergence to zero (as T→∞) of the variables $ \sup _{{0 \leqslant t \leqslant T - \alpha _{T} }} \inf _{{f \in S}} \sup _{{0 \leqslant x \leqslant 1}} {\left| {Y_{{t,T}} {\left( x \right)} - f{\left( x \right)}} \right|} Let {W(t); t≥ 0} be a standard Wiener process and S be the Strassen set of functions. We investigate the exact rates of convergence to zero (as T→∞) of the variables sup0≤
t
≤
T
−
aT
inf
f∈S
sup0≤
x
≤1|Y
t,T
(x) −f(x)| and inf0≤
t
≤
T−aT
sup0≤
x
≤1|Y
t,T
(x−f(x)| for any given f∈S, where Y
t,T
(x) = (W(t+xa
T
) −W(t)) (2a
T
(log Ta
T
−1 + log log T))−1/2.
We establish a relation between how small the increments are and the functional limit results of Cs?rg{\H o}-Révész increments
for a Wiener process. Similar results for partial sums of i.i.d. random variables are also given.
Received September 10, 1999, Accepted June 1, 2000 相似文献
10.
Luciana Angiuli Michele MirandaJr Diego Pallara Fabio Paronetto 《Annali di Matematica Pura ed Applicata》2009,188(2):297-331
Given a uniformly elliptic second order operator on a possibly unbounded domain , let (T(t))
t≥0 be the semigroup generated by in L
1(Ω), under homogeneous co-normal boundary conditions on ∂Ω. We show that the limit as t → 0 of the L
1-norm of the spatial gradient D
x
T(t)u
0 tends to the total variation of the initial datum u
0, and in particular is finite if and only if u
0 belongs to BV(Ω). This result is true also for weighted BV spaces. A further characterization of BV functions in terms of the short-time behaviour of (T(t))
t≥0 is also given.
相似文献
11.
Let {S
n
} be a random walk on ℤ
d
and let R
n
be the number of different points among 0, S
1,…, S
n
−1. We prove here that if d≥ 2, then ψ(x) := lim
n
→∞(−:1/n) logP{R
n
≥nx} exists for x≥ 0 and establish some convexity and monotonicity properties of ψ(x). The one-dimensional case will be treated in a separate paper.
We also prove a similar result for the Wiener sausage (with drift). Let B(t) be a d-dimensional Brownian motion with constant drift, and for a bounded set A⊂ℝ
d
let Λ
t
= Λ
t
(A) be the d-dimensional Lebesgue measure of the `sausage' ∪0≤
s
≤
t
(B(s) + A). Then φ(x) := lim
t→∞:
(−1/t) log P{Λ
t
≥tx exists for x≥ 0 and has similar properties as ψ.
Received: 20 April 2000 / Revised version: 1 September 2000 / Published online: 26 April 2001 相似文献
12.
Let M(σ) = sup{|F(σ + it)|: t ∈ ℝ} and μ(σ) = max {|a
n
|exp(σλn): n ≥ 0}, σ < 0, for a Dirichlet series {fx995-01} with abscissa of absolute convergence σa = 0. We prove that the condition ln ln n = o(ln λn), n → ∞, is necessary and sufficient for the equivalence of the relations {fx995-02}, for each series of this type.
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 851–856, June, 2008. 相似文献
13.
LetT(t) be the translation group onY=C
0(ℝ×K)=C
0(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY
⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y
⊙=L
1(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT
0(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT
0(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ
⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C
0(Ω;X) andL
p
(μ;X).
This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful
to the CWI and the Dutch National Science Foundation NWO for financial support. 相似文献
14.
The aim of this paper is to establish sufficient conditions of the finite time blow-up in solutions of the homogeneous Dirichlet
problem for the anisotropic parabolic equations with variable nonlinearity $
u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} }
$
u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} }
. Two different cases are studied. In the first case a
i
≡ a
i
(x), p
i
≡ 2, σ
i
≡ σ
i
(x, t), and b
i
(x, t) ≥ 0. We show that in this case every solution corresponding to a “large” initial function blows up in finite time if there
exists at least one j for which min σ
j
(x, t) > 2 and either b
j
> 0, or b
j
(x, t) ≥ 0 and Σπ
b
j
−ρ(t)(x, t) dx < ∞ with some σ(t) > 0 depending on σ
j
. In the case of the quasilinear equation with the exponents p
i
and σ
i
depending only on x, we show that the solutions may blow up if min σ
i
≥ max p
i
, b
i
≥ 0, and there exists at least one j for which min σ
j
> max p
j
and b
j
> 0. We extend these results to a semilinear equation with nonlocal forcing terms and quasilinear equations which combine
the absorption (b
i
≤ 0) and reaction terms. 相似文献
15.
MiaoLI QuanZHENG 《数学学报(英文版)》2004,20(5):821-828
Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups. 相似文献
16.
Yu Zhang 《Probability Theory and Related Fields》2006,136(2):298-320
We consider the first passage percolation model on Z
d
for d ≥ 2. In this model, we assign independently to each edge the value zero with probability p and the value one with probability 1−p. We denote by T(0, ν) the passage time from the origin to ν for ν ∈ R
d
and
It is well known that if p < p
c
, there exists a compact shape B
d
⊂ R
d
such that for all
> 0,
t
B
d
(1 −
) ⊂ B(t) ⊂ tB
d
(1 +
) and G(t)(1 −
) ⊂ B(t) ⊂ G(t)(1 +
) eventually w.p.1. We denote the fluctuations of B(t) from tB
d
and G(t) by
In this paper, we show that for all d ≥ 2 with a high probability, the fluctuations F(B(t), G(t)) and F(B(t), tB
d
) diverge with a rate of at least C log t for some constant C. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the
paths themselves. This linearity is also independently interesting.
Research supported by NSF grant DMS-0405150 相似文献
17.
Adam Bobrowski 《Journal of Evolution Equations》2007,7(3):555-565
Let
be a locally compact Hausdorff space. Let A and B be two generators of Feller semigroups in
with related Feller processes {X
A
(t), t ≥ 0} and {X
B
(t), t ≥ 0} and let α and β be two non-negative continuous functions on
with α + β = 1. Assume that the closure C of C
0 = αA + βB with
generates a Feller semigroup {T
C
(t), t ≥ 0} in
. It is natural to think of a related Feller process {X
C
(t), t ≥ 0} as that evolving according to the following heuristic rules. Conditional on being at a point
, with probability α(p) the process behaves like {X
A
(t), t ≥ 0} and with probability β(p) it behaves like {X
B
(t), t ≥ 0}. We provide an approximation of {T
C
(t), t ≥ 0} via a sequence of semigroups acting in
that supports this interpretation. This work is motivated by the recent model of stochastic gene expression due to Lipniacki
et al. [17]. 相似文献
18.
For uniformly stable bounded analytic C
0-semigroups {T(t)}
t≥0 of linear operators in a Banach space B, we study the behavior of their orbits T (t)x, x ∈ B, at infinity. We also analyze the relationship between the order of approaching the orbit T (t)x to zero as t → ∞ and the degree of smoothness of the vector x with respect to the operator A
−1 inverse to the generator A of the semigroup {T(t)}
t≥0. In particular, it is shown that, for this semigroup, there exist orbits approaching zero at infinity not slower than
, where a > 0, 0 < α < π/(2(π-θ)), θ is the angle of analyticity of {T(t)}
t≥0, and the collection of these orbits is dense in the set of all orbits.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 148–159, February, 2006. 相似文献
19.
Let X be a Banach space, A : D(A) X → X the generator of a compact C0- semigroup S(t) : X → X, t ≥ 0, D a locally closed subset in X, and f : (a, b) × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u'(t) = Au(t) + f(t, u(t - q)), t ∈ [to, to + T], with initial condition uto = φ ∈C([-q, 0]; X), is the tangency condition lim infh10 h^-1d(S(h)v(O)+hf(t, v(-q)); D) = 0 for almost every t ∈ (a, b) and every v ∈ C([-q, 0]; X) with v(0), v(-q)∈ D. 相似文献
20.
We investigate an initial value problem which is closely related to the Williams-Bjerknes tumour model for a cancer which
spreads through an epithelial basal layer modeled onI ⊂ Z
2. The solution of this problem is a familyp = (p
i(t)), where eachp
i(t)could be considered as an approximation to the probability that the cell situated ati is cancerous at timet. We prove that this problem has a unique solution, it is defined on [0, +∞[, and, for some relevant situations, limt→∞
P
i(t) = 1 for alli ∈ I. Moreover, we study the expected number of cancerous cells at timet. 相似文献