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1.
We consider the existence and uniqueness of singular solutions for equations of the formu
1=div(|Du|p−2
Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2.
Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the
existence of very singular solutions, i.e. singular solutions such that ∫|x|≤r
u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result.
In the case ϕ(u)=u
q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal.
Dedicated to Professor Shmuel Agmon 相似文献
2.
Huashui Zhan 《Applications of Mathematics》2008,53(6):521-533
We study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation u
t
= div(u
m−1|Du|
p−2
Du) − u
q
with an initial condition u(x, 0) = u
0(x). Here the exponents m, p and q satisfy m + p ⩾ 3, p > 1 and q > m + p − 2.
The paper was supported by NSF of China (10571144), NSF for youth of Fujian province in China (2005J037) and NSF of Jimei
University in China. 相似文献
3.
In this paper, we consider the global existence, uniqueness and L
∞ estimates of weak solutions to quasilinear parabolic equation of m-Laplacian type u
t
− div(|∇u|
m−2∇u) = u|u|
β−1 ∫Ω |u|
α
dx in Ω × (0,∞) with zero Dirichlet boundary condition in tdΩ. Further, we obtain the L
∞ estimate of the solution u(t) and ∇u(t) for t > 0 with the initial data u
0 ∈ L
q
(Ω) (q > 1), and the case α + β < m − 1. 相似文献
4.
D. S. Lubinsky 《Israel Journal of Mathematics》2006,153(1):193-219
In 1978, Freud, Giroux and Rahman established a weightedL
1 Jackson theorem for the weight exp(−|x|) on the real line, using methods that work only inL
1. This weight is somewhat exceptional, for it sits on the boundary between weights like exp(-|x|α), α≥1, where weighted polynomials are dense, and the case α<1, where weighted polynomials are not dense. We obtain the firstL
p
Jackson theorem for exp(−|x|), valid in allL
p
, 0<p≤∞, as well as for higher order moduli of continuity. We also establish a converse Bernstein type theorem, characterizing
rates of approximation in terms of smoothness of the approximated function.
Research supported by NSF grant DMS 0400446. 相似文献
5.
Maria E. Schonbek 《Mathematische Annalen》2006,336(3):505-538
This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u
t
−Δu+b(x)·∇(u|u|
q
−1)=f(x, t) in ℝ
n
×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u
0 is supposed to be in an appropriate L
p
space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant 相似文献
6.
Shu-Yu Hsu 《Mathematische Annalen》2003,325(4):665-693
We prove that the solution u of the equation u
t
=Δlog u, u>0, in (Ω\{x
0})×(0,T), Ω⊂ℝ2, has removable singularities at {x
0}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ0, C
1, C
2>0, such that C
1
|x−x
0|α≤u(x,t)≤C
2|x−x
0|−α holds for all 0<|x−x
0|≤ρ0 and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ2×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ2×(0,T) when 0≤u
0∉L
1
(ℝ2) is radially symmetric and u
0L
loc
1(ℝ2).
Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003
Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65 相似文献
7.
René L. Schilling 《Probability Theory and Related Fields》1998,112(4):565-611
Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C
c
∞(ℝ
n
)⊂D(A) and A|C
c
∞(ℝ
n
) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|∞≤c(1+|ξ|2) and |Imp(x,ξ)|≤c
0Rep(x,ξ). We show that the associated Feller process {X
t
}
t
≥0 on ℝ
n
is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour
of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β∞
x
:={λ>0:lim
|ξ|→∞
|
x
−
y
|≤2/|ξ||p(y,ξ)|/|ξ|λ=0} or δ∞
x
:={λ>0:liminf
|ξ|→∞
|
x
−
y
|≤2/|ξ|
|ε|≤1|p(y,|ξ|ε)|/|ξ|λ=0}, and obtain a.s. (ℙ
x
) that lim
t
→0
t
−1/λ
s
≤
t
|X
s
−x|=0 or ∞ according to λ>β∞
x
or λ<δ∞
x
. Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27].
Received: 21 July 1997 / Revised version: 26 January 1998 相似文献
8.
Baode Li Yinsheng Jiang Hui Cao 《分析论及其应用》2007,23(2):138-146
In the paper we obtain vector-valued inequalities for Calderon-Zygmund operator,simply CZO On Herz space and weak Herz space.In particular,we obtain vector-valued inequalities for CZO on Lq(Rd,│x│αdμ)space,with 1<q<∞,-n<α<n(q-1),and on L1,∞(Rd,│x│αdμ)space,with -n<α<0. 相似文献
9.
The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves ei ω tϕω(x) for a nonlinear Schr?dinger equation with an inhomogeneous nonlinearity V (x)|u|p − 1u, where V (x) is proportional to the electron density. Here, ω > 0 and ϕω(x) is a ground state of the stationary problem. When V (x) behaves like |x|−b at infinity, where 0 < b < 2, we show that ei ω tϕω(x) is stable for p < 1 + (4 − 2b)/n and sufficiently small ω > 0. The main point of this paper is to analyze the linearized operator at standing wave solution
for the case of V (x) = |x|−b. Then, this analysis yields a stability result for the case of more general, inhomogeneous V (x) by a certain perturbation method.
Communicated by Bernard Helffer
submitted 14/07/04, accepted 28/02/05 相似文献
10.
R. Ya. Glagoleva 《Mathematical Notes》1996,60(6):622-628
We consider the solutions of degenerate parabolic equations and inequalities of the formLu-u
t
= |u|
q
sgnu and sgnu(Lu−u
t
)−|u|
q
≥0, 0<q<1, with the elliptic operatorL in divergent or nondivergent form. We establish a dependence of the maximum modulus of the solution on the domain and on
the equation (inequality) such that this dependence guarantees the existence of a “dead zone” of the solution. In this case,
the character of degeneracy is unessential.
Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 824–831, December, 1996. 相似文献
11.
We obtain the best approximation in L
1(ℝ), by entire functions of exponential type, for a class of even functions that includes e
−λ|x|, where λ>0, log |x| and |x|
α
, where −1<α<1. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded
degree. 相似文献
12.
Katalin Marton 《Probability Theory and Related Fields》1998,110(3):427-439
Summary. Let X={X
i
}
i
=−∞
∞ be a stationary random process with a countable alphabet and distribution q. Let q
∞(·|x
−
k
0) denote the conditional distribution of X
∞=(X
1,X
2,…,X
n
,…) given the k-length past:
Write d(1,x
1)=0 if 1=x
1, and d(1,x
1)=1 otherwise. We say that the process X admits a joining with finite distance u if for any two past sequences −
k
0=(−
k
+1,…,0) and x
−
k
0=(x
−
k
+1,…,x
0), there is a joining of q
∞(·|−
k
0) and q
∞(·|x
−
k
0), say dist(0
∞,X
0
∞|−
k
0,x
−
k
0), such that
The main result of this paper is the following inequality for processes that admit a joining with finite distance:
Received: 6 May 1996 / In revised form: 29 September 1997 相似文献
13.
Let S ⊂ ℜn+1
be the graph of the function ϕ :[−1, 1]
n → ℜ defined by ϕ (x
1
, …, xn) = ∑
j=1
n
|xj|αj, with1<α
1
≤ … ≤ αn, let σ the Euclidean area measure on S. In this article we study the Lp − Lq boundedness of convolution operators with the singular Borel measure on Rn+1
given by μ (E)=σ (E ∩ S) 相似文献
14.
Abstract. It is proved that the semilinear elliptic problem with zero boundary value 相似文献
15.
Piero D’Ancona Vittoria Pierfelice Fulvio Ricci 《Journal of Fourier Analysis and Applications》2010,16(2):294-310
The dispersive properties of the wave equation u
tt
+Au=0 are considered, where A is either the Hermite operator −Δ+|x|2 or the twisted Laplacian −(∇
x
−iy)2/2−(∇
y
+ix)2/2. In both cases we prove optimal L
1−L
∞ dispersive estimates. More generally, we give some partial results concerning the flows exp (itL
ν
) associated to fractional powers of the twisted Laplacian for 0<ν<1. 相似文献
16.
In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative
scheme for the equation (ϕ
p
(u′))′+q(t)f(u) = 0, 0 < t < 1, where ϕ
p
(s):= |s|
p−2
s, p > 1, subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient q(t) may be singular at t = 0; 1. 相似文献
17.
Abstract
We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding an asymptotic formula for the number of primitive λ-roots in any interval I of length ∣I∣ ≫ q
1/2+∈. Similar results are obtained for the distribution of ordered pairs (x, x
−1) with x a primitive λ-root, and for the number of primitive λ-roots satisfying inequalities such as |x − x
−1| ≤ B.
(Dedicated to Professor Wang Yuan on the occasion of his 75th birthday)
*Project supported by the National Natural Science Foundation of China (No.19625102) and the 973 Project of the Ministry of
Science and Technology of China. 相似文献
18.
Spaces of analytic functions of Hardy-Bloch type 总被引:1,自引:1,他引:0
For 0<p≤∞ and 0<q≤∞, the space of Hardy-Bloch type ℬ(p,q) consists of those functionsf which are analytic in the unit diskD such that (1−r)M
p
(r,f′)⊂L
q
(dr/(1−r)). We note that ℬ(∞,∞) coincides with the Bloch space ℬ and that ℬ⊂ℬ(p,∞) for allp. Also, the space ℬ(p,p) is the Dirichlet spaceD
p−1
p
.
We prove a number of results on decomposition of spaces with logarithmic weights which allow us to obtain sharp results about
the mean growth of the ℬ(p,q). In particular, we prove that iff is an analytic function inD and 2≤p<∞, then the conditionM
p
(r,f′)=O((1−r)−1), asr→1, implies that
. This result is an improvement of the well-known estimate of Clunie and MacGregor and Makarov about the integral means of
Bloch functions, and it also improves the main result in a recent paper by Girela and Peláez. We also consider the question
of characterizing the univalent functions in the spaces ℬ(p,2), 0<p<∞, and in some other related spaces and give some applications of our estimates to study the Carleson measures for the spaces
ℬ(p,2) andD
p−1
p
.
The first and third authors were supported by grants from “E1 Ministerio de Educación y Ciencia”, Spain (MTN2004-00078 and
MTN2004-21420-E) and by a grant from “La Junta de Andalucía” (FQM-210).
The second author was supported in part by MNZŽS Grant, No. ON144010, Serbia. 相似文献
19.
Pascale Vitse 《Rendiconti del Circolo Matematico di Palermo》2004,53(2):283-312
For Banach space operatorsT satisfying the Tadmor-Ritt condition ‖(zI−T)−1‖≤C|z−1|−1, |z|>1, we show how to use the Riesz turndown collar theorem to estimate sup
n≥0‖T
n‖. A similar estimate is shown for lim sup
n
‖T
n‖ in terms of the Ritt constantM=lim sup
z→1‖(1−z)(zI−T)−1‖. We also obtain an estimate of the functional calculus for these operators proving, in particular, that ‖f(T)‖≤C
q‖f‖
Mult
, where ‖·‖
Mult
stands for the multiplier norm of the Cauchy-Stieltjes integrals over a Lusin type cone domain depending onC and a parameterq, 0<q<1.
Notation.D denotes the open unit disc of the complex plane,D={z∈ℂ:|z|<1}, andT={z∈ℂ:|z|=1} is the unit circle.H
∞ is the Banach algebra of bounded analytic functions onD equipped with the supremum norm ‖.‖∞. 相似文献
20.
Ignacy I. Kotlarski 《Annali di Matematica Pura ed Applicata》1969,83(1):253-260
Summary The probability density functions fk(xk)=Ak|xk|p
k−1 e−aφ
k(xk) of independent random variables x0, x1, ..., xn, are characterized by independence of two functions of them.
Entrata in Redazione il 12 aprile 1969. 相似文献