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1.
E. M. E. Zayed 《Applicable analysis》2013,92(9):905-925
The asymptotic expansions of the trace of the heat kernel for small positive t, where λν are the eigenvalues of the negative Laplacian in Rn (n=2 or 3), are studied for a general annular bounded domain Ω with a smooth inner boundary ?Ω1 and a smooth outer boundary ?Ω2 where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the components Γ j (j=1,…,m) of ?Ω1 and on the components of ?Ω2 are considered such that and and where the coefficients in the Robin boundary conditions are piecewise smooth positive functions. Some applications of Θ (t) for an ideal gas enclosed in the general annular bounded domain Ω are given. 相似文献
2.
《Applicable analysis》2013,92(5):401-409
In this article, we obtain some results on the linearized oscillation of the odd-order neutral difference equation <artwork name="GAPA31013eu1"> where Δ is the forward difference operator, m is odd, {pn },{qn } are sequences of nonnegative real numbers, k, l are nonnegative integers, g(x),h(x)∈ C(R,R) with xg(x) > 0 for x ≠0. 相似文献
3.
Thomas H. Pate 《Linear and Multilinear Algebra》2013,61(4):361-391
Given a <artwork name="GLMA31007ei1">-valued function f with domain <artwork name="GLMA31007ei2">, the symmetric group on {1,2,…, m}, we define the generalized matrix function [ f ](?), or df (?), in the usual way on the set of all m× m complex matrices. Letting <artwork name="GLMA31007ei3"> denote the set of all m× m positive semi-definite Hermitian matrices we consider the cone K m whose elements are the Hermitian functions <artwork name="GLMA31007ei4"> such that [ f ]( A)≥0 for all <artwork name="GLMA31007ei5">. The extreme rays in K m are fundamental to an understanding of the linear inequalities that result by restricting the generalized matrix functions [ f ](?) to the sets <artwork name="GLMA31007ei6">. In particular, the resolution of Lieb's permanent dominance conjecture, and certain similar conjectures such as the conjecture of Soules, will likely require identification and careful analysis of these rays. Grone, Merris, and Watkins have shown that the determinant function det(?), which is [ f ](?) if f is the signum function, is extreme in K m for each m. We identify additional rays that are extreme for all m. In particular, we associate with each 2-term partition <artwork name="GLMA31007ei7"> of {1,2,…, m} an element <artwork name="GLMA31007ei8"> that is shown to be extreme in K m for each m. If <artwork name="GLMA31007ei9"> is trivial, then <artwork name="GLMA31007ei10"> reduces to the determinant function; hence, our results are a natural extension of the result of Grone, Merris, and Watkins. Moreover <artwork name="GLMA31007ei11">, like det ( A), is expressible as a function of the eigenvalues of certain matrices related to A. Additional classes of extreme rays are also presented. 相似文献
4.
P. Quittner W. Reichel 《Calculus of Variations and Partial Differential Equations》2008,32(4):429-452
Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂ν
u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L
∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s|
p
) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that
p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) = s
p
then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of
∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential
equation is of the form h(x, u) with h satisfying suitable growth conditions. 相似文献
5.
We build a version of a thermodynamic formalism for maps
of the form f(z) = ∑
j = 0
p + q
a
j
e
(j−p)z
where p, q > 0 and
. We show in particular the existence and uniqueness of (t,α)-conformal measures and that the Hausdorff dimension HD(J
f
r
) = h is the unique zero of the pressure function t ↦ P(t) for t > 1, where the set J
f
r
is the radial Julia set.
Partially supported by NSF Grant DMS 0100078.
Partially supported by Warsaw University of Technology Grant No. 504G11200023000, Polish KBN Grant No. 2PO3A03425 and Chilean
FONDECYT Grant No. 11060280. 相似文献
6.
In this paper we shall consider the critical elliptic
equation
where
and a(x)
is a real continuous, non
negative function, not identically zero. By using a local Pohozaev
identity, we show that problem (0.1) does not admit a
family of solutions
which blows-up and concentrates as
at some zero point x0 of a(x)
if the order of flatness of the function a(x) at x0 is
相似文献
7.
Huan-song Zhou Hong-bo Zhu 《应用数学学报(英文版)》2007,23(4):685-696
In this paper,we consider the following ODE problem(P)where f ∈ C((0, ∞)×R,R),f(r,s)goes to p(r)and q(r)uniformly in r>0 as s→0 and s→ ∞,respectively,0≤p(r)≤q(r)∈ L~∞(0,∞).Moreover,for r>0,f(r,s)is nondecreasing in s≥0.Some existenceand non-existence of positive solutions to problem(P)are proved without assuming that p(r)≡0 and q(r)hasa limit at infinity.Based on these results,we get the existence of positive solutions for an elliptic problem. 相似文献
8.
In this paper we consider the elliptic system Δu=a(x)upvq, Δv=b(x)urvs in Ω, a smooth bounded domain, with boundary conditions , on ∂Ω. Here λ and μ are regarded as parameters and p,s>1, q,r>0 verify (p−1)(s−1)>qr. We consider the case where a(x)?0 in Ω and a(x) is allowed to vanish in an interior subdomain Ω0, while b(x)>0 in . Our main results include existence of nonnegative nontrivial solutions in the range 0<λ<λ1?∞, μ>0, where λ1 is characterized by means of an eigenvalue problem, and the uniqueness of such solutions. We also study their asymptotic behavior in all possible cases: as both λ,μ→0, as λ→λ1<∞ for fixed μ (respectively μ→∞ for fixed λ) and when both λ,μ→∞ in case λ1=∞. 相似文献
9.
In this paper,the L2-boundedness of a class of parametric Marcinkiewicz integral μρΩ,h with kernel function Ω in Bq0.0 (Sn-1) for some q> 1,and the radial function h (x)∈ l∞ (Ls) (R+) for 1<s≤∞ are given. The Lp(Rn) (2≤p<∞) boundedness of μ*Ω,ph,λ and μρΩ,h,s with Ω in Bq0,0(Sn-1) and h(|x|)∈l∞(Ls)(R+) in application are obtained. Here μ*Ω,p h,λ and μpΩ,h,s are parametric Marcinkiewicz integrals corresponding to the Littlewood-Paley gλ* function and the Lusin area function S,respectively. 相似文献
10.
This article gives a simple proof of an equivalent proposition on semiconcave function (see [L.C. Evans (1998). Partial Differential Equations. American Mathematical Society; p. 130]). The proof of sufficiency of the proposition can be easily obtained. We prove its necessity by three steps: First, we prove that the equivalent proposition holds for discrete points <artwork name="GAPA31045ei1">; Secondly, we obtain continuity of semiconcave function; Finally, by using the fact that the sequences λm k are dense in the interval (0, 1), we prove that the equivalent proposition holds for each λ ∈ (0, 1). 相似文献
11.
Goro Akagi 《Journal of Differential Equations》2007,241(2):359-385
The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|p−2∇u), with initial data u0∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u0 and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T0] in which the problem admits a solution. More precisely, T0 depends only on Lr|u0| and f. 相似文献
12.
Flávio Dickstein 《Journal of Differential Equations》2006,223(2):303-328
We study the Cauchy problem for the nonlinear heat equation ut-?u=|u|p-1u in RN. The initial data is of the form u0=λ?, where ?∈C0(RN) is fixed and λ>0. We first take 1<p<pf, where pf is the Fujita critical exponent, and ?∈C0(RN)∩L1(RN) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<ps, where ps is the Sobolev critical exponent, and ?(x) decaying as |x|-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial. 相似文献
13.
Vincenzo De Filippis 《Israel Journal of Mathematics》2009,171(1):325-348
Let R be a prime ring with extended centroid C, δ a nonzero generalized derivation of R, f(x
1, ..., x
n
) a nonzero multilinear polynomial over C, I a nonzero right ideal of R and k ≥ a fixed integer.
If [δ(f(r
1, ..., r
n
)), f(r
1, ..., r
n
)]
k
= 0, for all r
1, ..., r
n
∈ I, then either δ(x) = ax, with (a-γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds
(1) if char(R) = 0 then f(x
1, ..., x
n
) is central valued in eRCe
(2) if char(R) = p > 0 then is central valued in eRCe, for a suitable s ≥ 0, unless when char(R) = 2 and eRCe satisfies the standard identity s
4
(3) δ(x) = ax−xb, where (a+b+α)e = 0, for α ∈ C, and f(x
1, ..., x
n
)2 is central valued in eRCe. 相似文献
14.
Zongming Guo 《Mathematische Nachrichten》2002,243(1):43-74
The structure of positive solutions to the quasilinear elliptic problems –div(|Du|p–2Du = λf(u) in Ω, u = 0 on ∂Ω, p > 1, Ω ⊂ RNa bounded smooth domain, is precisely studied when λ is sufficiently large, for a class of logistic‐type nonlinearities f(u) satisfying that f(0) = f(a) = 0, a > 0, f(u) > 0 for u ∈ (0,a), , while u = a is a zero point of f with order ω. It is shown that if ω ≥ p – 1, the problem has a unique positive solution uλ with sup Ω uλ < a, which develops a boundary layer near ∂Ω. It is shown that if 0 < ω < p – 1, the problem also has a unique positive solution u λ, but the flat core {x ∈ Ω : uλ(x) = a} ≠ ∅︁ exists. Moreover, the asymptotic behaviour of the flat core is studied as λ → ∞. 相似文献
15.
Zhu Ning 《高校应用数学学报(英文版)》1998,13(3):241-250
ANOTEONTHEBEHAVIOROFBLOW┐UPSOLUTIONSFORONE┐PHASESTEFANPROBLEMSZHUNINGAbstract.Inthispaper,thefolowingone-phaseStefanproblemis... 相似文献
16.
Evangelos A. Latos Dimitrios E. Tzanetis 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):137-151
We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ut = (un)xx + lf(u)/(ò-11 f(u)dx)2{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s
n-1
f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution.
For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}. 相似文献
17.
Olof J Staffans 《Journal of Mathematical Analysis and Applications》1985,108(1):15-30
Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝0ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝0tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝0ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f ∝0tr(t, s) f(s) ds maps a weighted L1 space continuously into another weighted L1 space, and a weighted L∞ space into another weighted L∞ space. Our main theorem is used to study the asymptotic behavior of two differential delay equations. 相似文献
18.
We consider the following singularly perturbed boundary-value problem:
on the interval 0 ≤x ≤ 1. We study the existence and uniqueness of its solutionu(x, ε) having the following properties:u(x, ε) →u
0(x) asε → 0 uniformly inx ε [0, 1], whereu
0(x) εC
∞ [0, 1] is a solution of the degenerate equationf(x, u, u′)=0; there exists a pointx
0 ε (0, 1) such thata(x
0)=0,a′(x
0) > 0,a(x) < 0 for 0 ≤x <x
0, anda(x) > 0 forx
0 <x ≤ 1, wherea(x)=f′
v(x,u
0(x),u′
0(x)).
Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 520–524, April, 2000. 相似文献
19.
Explosive solutions of elliptic equations with absorption and nonlinear gradient term 总被引:2,自引:0,他引:2
Marius Ghergu Constantin Niculescu Vicenţiu Rădulescu 《Proceedings Mathematical Sciences》2002,112(3):441-451
Letf be a non-decreasing C1-function such that
andF(t)/f
2
a(t)→ 0 ast → ∞, whereF(t)=∫
0
t
f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu|
a
=p(x)f(u) in a smooth bounded domain Ω ⊂RN, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded. 相似文献
20.
We derive the approximation on [0, 1] of functionsf(x) by interpolating spline-functions sr(f; x) of degree 2r+1 and defect r+1 (r=1, 2,...). Exact estimates for ¦f(x)–sr(f; x) ¦ and f(x)–sr(f; x)|c on the class WmH for m=1, r=1, 2, ..., and m=2, 3, r=1 for the case of convex (t),are derived.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 483–494, May, 1971. 相似文献