共查询到20条相似文献,搜索用时 31 毫秒
1.
We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with +(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u0∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u0(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L1-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L∞‖ for (2+mn)/(n+1)<p<2. 相似文献
2.
N. G. Aharonyan V. K. Ohanyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2018,53(2):113-120
In the paper, a formula to calculate the probability that a random segment L(ω, u) in R n with a fixed direction u and length l lies entirely in the bounded convex body D ? R n (n ≥ 2) is obtained in terms of covariogram of the body D. For any dimension n ≥ 2, a relationship between the probability P(L(ω, u) ? D) and the orientation-dependent chord length distribution is also obtained. Using this formula, we obtain the explicit form of the probability P(L(ω, u) ? D) in the cases where D is an n-dimensional ball (n ≥ 2), or a regular triangle on the plane. 相似文献
3.
D.L. Burkholder 《Journal of Functional Analysis》1975,18(4):429-454
Let N be the nontangential maximal function of a function u harmonic in the Euclidean half-space Rn × (0, ∞) and let N? be the nontangential maximal function of its negative part. If u(0, y) = o(y?n) as y → ∞, then ∥N∥p ? cp ∥N?∥p, 0 < p < 1, and more. The basic inequality of the paper (Theor. 2.1) can be used not only to derive such global results but also may be used to study the behavior of u near the boundary. Similar results hold for martingales with continuous sample functions. In addition, Theorem 1.3 contains information about the zeros of u. For example, if u belongs to Hp for some 0 < p < 1, then every thick cone in the half-space must contain a zero of u. 相似文献
4.
Let w(x) = (1 - x)α (1 + x)β be a Jacobi weight on the interval [-1, 1] and 1 < p < ∞. If either α > ?1/2 or β > ?1/2 and p is an endpoint of the interval of mean convergence of the associated Fourier-Jacobi series, we show that the partial sum operators Sn are uniformly bounded from Lp,1 to Lp,∞, thus extending a previous result for the case that both α, β > ?1/2. For α, β > ?1/2, we study the weak and restricted weak (p, p)-type of the weighted operators f→uSn(u?1f), where u is also Jacobi weight. 相似文献
5.
David M. Mason 《Stochastic Processes and their Applications》1983,15(1):99-109
Let Gn denote the empirical distribution based on n independent uniform (0, 1) random variables. The asymptotic distribution of the supremum of weighted discrepancies between Gn(u) and u of the forms 6wv(u)Dn(u)6 and 6wv(Gn(u))Dn(u)6, where Dn(u) = Gn(u)?u, wv(u) = (u(1?u))?1+v and 0 ? v < is obtained. Goodness-of-fit tests based on these statistics are shown to be asymptotically sensitive only in the extreme tails of a distribution, which is exactly where such statistics that use a weight function wv with ? v ? 1 are insensitive. For this reason weighted discrepancies which use the weight function wv with 0 ? v < are potentially applicable in the construction of confidence contours for the extreme tails of a distribution. 相似文献
6.
R.S. Singh 《Journal of multivariate analysis》1976,6(2):338-342
Let Xj = (X1j ,…, Xpj), j = 1,…, n be n independent random vectors. For x = (x1 ,…, xp) in Rp and for α in [0, 1], let Fj(x) = αI(X1j < x1 ,…, Xpj < xp) + (1 ? α) I(X1j ≤ x1 ,…, Xpj ≤ xp), where I(A) is the indicator random variable of the event A. Let Fj(x) = E(Fj(x)) and Dn = supx, α max1 ≤ N ≤ n |Σ0n(Fj(x) ? Fj(x))|. It is shown that P[Dn ≥ L] < 4pL exp{?2(L2n?1 ? 1)} for each positive integer n and for all L2 ≥ n; and, as n → ∞, with probability one. 相似文献
7.
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
8.
José María Martell 《Journal of Mathematical Analysis and Applications》2004,294(1):223-236
We prove two-weight, weak type norm inequalities for potential operators and fractional integrals defined on spaces of homogeneous type. We show that the operators in question are bounded from Lp(v) to Lq,∞(u), 1<p?q<∞, provided the pair of weights (u,v) verifies a Muckenhoupt condition with a “power-bump” on the weight u. 相似文献
9.
For 0 < m < n, p a positive integer and p > n/(n ? m), we study the inhomogeneous equation L u +u p + V (x)u + f(x) = 0 in ? n with singular data f and V. The symbol σ of the operator L is bounded from below by |ξ| m . Examples of L are Laplacian, biharmonic and fractional order operators. Here f and V can have infinite singular points, change sign, oscillate at infinity, and be measures. Also, f and V can blow up on an unbounded (n?1)-manifold. The solution u can change sign, be nonradial and singular. If σ, f and V are radial, then u is radial. The assumptions on f and V are in terms of their Fourier transforms and we provide some examples. 相似文献
10.
Najla A. Altwaijry 《Journal of Functional Analysis》2008,254(11):2866-2892
The Banach-Lie algebra L(A) of multiplication operators on the JB∗-triple A is introduced and it is shown that the hermitian part Lh(A) of L(A) is a unital GM-space the base of the dual cone in the dual GL-space ∗(Lh(A)) of which is affine isomorphic and weak∗-homeomorphic to the state space of L(A). In the case in which A is a JBW∗-triple, it is shown that tripotents u and v in A are orthogonal if and only if the corresponding multiplication operators in the unital GM-space Lh(A) satisfy
0?D(u,u)+D(v,v)?idA, 相似文献
11.
Frederick Bloom 《Journal of Mathematical Analysis and Applications》1977,59(3):469-487
For the abstract Volterra integro-differential equation utt ? Nu + ∝?∞t K(t ? τ) u(τ) dτ = 0 in Hilbert space, with prescribed past history u(τ) = U(τ), ? ∞ < τ < 0, and associated initial data u(0) = f, ut(0) = g, we establish conditions on K(t), ? ∞ < t < + ∞ which yield various growth estimates for solutions u(t), belonging to a certain uniformly bounded class, as well as lower bounds for the rate of decay of solutions. Our results are interpreted in terms of solutions to a class of initial-boundary value problems in isothermal linear viscoelasticity. 相似文献
12.
Maria De Natividade 《Monatshefte für Mathematik》2011,164(1):87-114
Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces L ??(w) in terms of the fundamental function of L ??(w). In particular, we prove that these bases are greedy in L ??(w) if and only if L ??(w) =?L p (w), 1?<?p?<???. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For L p (w) the approximation spaces are identified with weighted Besov spaces. 相似文献
13.
H.O Cordes 《Journal of Functional Analysis》1975,18(2):115-131
Commutators [a(M), b(D)] of a multiplication (a(M)u)(x) = a(x) u(x) and a convolution b(D) = F?1b(M)F (F = Fourier transform) are L2-compact if only the continuous functions a and b are bounded and for c = a and c = b we have . An improvement of a result by Calderon and Vaillancourt of boundedness of pseudodifferential operators is discussed (including an independent proof). Similar results on Lp-compactness and Lp-boundedness, 1 < p < ∞, using the Hoermander-Mihlin boundedness theorem on n-Fourier-multipliers, and with conditions and proofs different from the case of L2. 相似文献
14.
Our aim in this note is to deal with boundary limits of monotone Sobolev functions with ▽u∈ Lp(·)logLq(·)(B) for the unit ball BRn. Here p(·) and q(·) are variable exponents satisfyingthe log-Hlder and the log log-Hlder conditions, respectively. 相似文献
15.
Let D be a positive integer, and let p be an odd prime with p ? D. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M.A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for N(D, p), and also prove that if the equation U 2 ? DV 2 = ?1 has integer solutions (U, V), the least solution (u 1, v 1) of the equation u 2 ? pv 2 = 1 satisfies p ? v 1, and D > C(p), where C(p) is an effectively computable constant only depending on p, then the equation x 2 ? D = p n has at most two positive integer solutions (x, n). In particular, we have C(3) = 107. 相似文献
16.
Lucas C.F. Ferreira 《Journal of Differential Equations》2011,250(4):2045-2063
We study the equation Δu+u|u|p−1+V(x)u+f(x)=0 in Rn, where n?3 and p>n/(n−2). The forcing term f and the potential V can be singular at zero, change sign and decay polynomially at infinity. We can consider anisotropic potentials of form h(x)|x|−2 where h is not purely angular. We obtain solutions u which blow up at the origin and do not belong to any Lebesgue space Lr. Also, u is positive and radial, in case f and V are. Asymptotic stability properties of solutions, their behavior near the singularity, and decay are addressed. 相似文献
17.
Kazushi Yoshitomi 《Indagationes Mathematicae》2005,16(2):289-299
Let q ∈ {2, 3} and let 0 = s0 < s1 < … < sq = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set lj = sj − sj−1 for j ∈ 1, q. Given (p1,, pq) ∈ Rq, let b: Z → R be a periodic function of period T such that b(·) = pj on sj−1 + 1, sj for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l2(Z). By [λ2j , λ2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that pm ≠ pn for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1. 相似文献
18.
Hans Engler 《Journal of Functional Analysis》1985,64(3):412-435
Consider the heat equation ?ru ? Δxu = 0 in a cylinder Ω × [0,T] ? n+1 smooth lateral boundary under zero Neumann or Dirichlet conditions. Geometric conditions for Ω are given that guarantee that for a given P, 6▽xu(·, t)6Lp will be non-increasing for any solution. Decay rates are also given. For arbitrary Ω and p, it is shown how to construct an equivalent Lp-norm, such that ▽x(·, t) is non-increasing in this norm. 相似文献
19.
Let u =(uh, u3) be a smooth solution of the 3-D Navier-Stokes equations in R3× [0, T). It was proved that if u3 ∈ L∞(0, T;˙B-1+3/p p,q(R3)) for 3 p, q ∞ and uh∈ L∞(0, T; BMO-1(R3)) with uh(T) ∈ VMO-1(R3), then u can be extended beyond T. This result generalizes the recent result proved by Gallagher et al.(2016), which requires u ∈ L∞(0, T;˙B-1+3/pp,q(R3)). Our proof is based on a new interior regularity criterion in terms of one velocity component, which is independent of interest. 相似文献
20.
Mohamed Benrhouma 《Journal of Mathematical Analysis and Applications》2009,358(2):307-2694
In this paper, we study the existence and the uniqueness of positive solution for the sublinear elliptic equation, −Δu+u=p|u|sgn(u)+f in RN, N?3, 0<p<1, f∈L2(RN), f>0 a.e. in RN. We show by applying a minimizing method on the Nehari manifold that this problem has a unique positive solution in H1(RN)∩Lp+1(RN). We study its continuity in the perturbation parameter f at 0. 相似文献