共查询到20条相似文献,搜索用时 15 毫秒
1.
By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term h affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x), u>0 in Ω, u|∂Ω=∞, where Ω is a bounded domain with smooth boundary in RN, λ>0, g∈C1[0,∞) is increasing on [0,∞), g(0)=0, g′ is regularly varying at infinity with positive index ρ, the weight b, which is non-trivial and non-negative in Ω, may be vanishing on the boundary, and the inhomogeneous term h is non-negative in Ω and may be singular on the boundary. 相似文献
2.
The paper deals with the radially symmetric solutions of ut=Δu+um(x,t)vn(0,t), vt=Δv+up(0,t)vq(x,t), subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that u (v) blows up alone if and only if m>p+1 (q>n+1), which means that any blow-up is simultaneous if and only if m≤p+1, q≤n+1. (ii) Any blow-up is u (v) blowing up with v (u) remaining bounded if and only if m>p+1, q≤n+1 (m≤p+1, q>n+1). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1, q>n+1. Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model. 相似文献
3.
We consider the semilinear parabolic equation ut=Δu+up on RN, where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all x∈RN and t∈R. Our main result asserts that there are no positive radially symmetric bounded entire solutions. Then we consider radial solutions of the Cauchy problem. We show that if such a solution is global, that is, defined for all t?0, then it necessarily converges to 0, as t→∞, uniformly with respect to x∈RN. 相似文献
4.
Suppose X is a real q-uniformly smooth Banach space and F,K:X→X are Lipschitz ?-strongly accretive maps with D(K)=F(X)=X. Let u∗ denote the unique solution of the Hammerstein equation u+KFu=0. An iteration process recently introduced by Chidume and Zegeye is shown to converge strongly to u∗. No invertibility assumption is imposed on K and the operators K and F need not be defined on compact subsets of X. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included. 相似文献
5.
This paper is devoted to construct a family of fifth degree cubature formulae for n-cube with symmetric measure and n-dimensional spherically symmetrical region. The formula forn-cube contains at most n2+5n+3 points and for n-dimensional spherically symmetrical region contains only n2+3n+3 points. Moreover, the numbers can be reduced to n2+3n+1 and n2+n+1 if n=7 respectively, the latter of which is minimal. 相似文献
6.
Let r,s∈]1,2[ and λ,μ∈]0,+∞[. In this paper, we deal with the existence and multiplicity of nonnegative and nonzero solutions of the Dirichlet problem with 0 boundary data for the semilinear elliptic equation −Δu=λus−1−ur−1 in Ω⊂RN, where N≥2. We prove that there exists a positive constant Λ such that the above problem has at least two solutions, at least one solution or no solution according to whether λ>Λ, λ=Λ or λ<Λ. In particular, a result by Hernandéz, Macebo and Vega is improved and, for the semilinear case, a result by Díaz and Hernandéz is partially extended to higher dimensions. Finally, an answer to a conjecture, recently stated by the author, is also given. 相似文献
7.
Michel Mandjes Petteri Mannersalo Ilkka Norros Miranda van Uitert 《Stochastic Processes and their Applications》2006
Consider events of the form {Zs≥ζ(s),s∈S}, where Z is a continuous Gaussian process with stationary increments, ζ is a function that belongs to the reproducing kernel Hilbert space R of process Z, and S⊂R is compact. The main problem considered in this paper is identifying the function β∗∈R satisfying β∗(s)≥ζ(s) on S and having minimal R-norm. The smoothness (mean square differentiability) of Z turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ(s)=s for s∈[0,1] and Z is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process. 相似文献
8.
Suppose X is a real q-uniformly smooth Banach space and F,K:X→X are bounded strongly accretive maps with D(K)=F(X)=X. Let u∗ denote the unique solution of the Hammerstein equation u+KFu=0. A new explicit coupled iteration process is shown to converge strongly to u∗. No invertibility assumption is imposed on K and the operators K and F need not be defined on compact subsets of X. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included. 相似文献
9.
In 2011, the fundamental gap conjecture for Schrödinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrödinger equation with a convex potential and relative error ε. Classical deterministic algorithms solving this problem have cost exponential in the number of its degrees of freedom d. We show a quantum algorithm, that is based on a perturbation method, for estimating the ground state energy with relative error ε. The cost of the algorithm is polynomial in d and ε−1, while the number of qubits is polynomial in d and logε−1. In addition, we present an algorithm for preparing a quantum state that overlaps within 1−δ,δ∈(0,1), with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error ε. The cost of the algorithm is polynomial in d, ε−1 and δ−1, while the number of qubits is polynomial in d, logε−1 and logδ−1. 相似文献
10.
Let H be a real Hilbert space. Let K,F:H→H be bounded, continuous and monotone mappings. Suppose that u∗∈H is a solution to the Hammerstein equation u+KFu=0. We construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the Hammerstein equation. Furthermore, we give some examples to show that our result is interdisciplinary in nature, covers a large variety of areas and should be of much interest to a wide audience. 相似文献
11.
We study the problem (−Δ)su=λeu in a bounded domain Ω⊂Rn, where λ is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7 for all s∈(0,1) whenever Ω is, for every i=1,...,n, convex in the xi-direction and symmetric with respect to {xi=0}. The same holds if n=8 and s?0.28206..., or if n=9 and s?0.63237.... These results are new even in the unit ball Ω=B1. 相似文献
12.
In this paper, we prove a comparison result between a solution u(x,t), x∈Ω⊂R2, t∈(0,T), of a time depending equation involving the Monge–Ampère operator in the plane and the solution of a conveniently symmetrized parabolic equation. To this aim, we prove a derivation formula for the integral of a smooth function g(x,t) over sublevel sets of u, {x∈Ω:u(x,t)<?}, ?∈R, having the same perimeter in R2. 相似文献
13.
The author deals with the quasilinear parabolic equation ut=[uα+g(u)]Δu+buα+1+f(u,∇u) with Dirichlet boundary conditions in a bounded domain Ω, where f and g are lower-order terms. He shows that, under suitable conditions on f and g, whether the solution is bounded or blows up in a finite time depends only on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For some special cases, the result is sharp. 相似文献
14.
Tertuliano Franco Patrícia Gonçalves Adriana Neumann 《Stochastic Processes and their Applications》2013
We analyze the equilibrium fluctuations of density, current and tagged particle in symmetric exclusion with a slow bond. The system evolves in the one-dimensional lattice and the jump rate is everywhere equal to one except at the slow bond where it is αn−β, with α>0, β∈[0,+∞] and n is the scaling parameter. Depending on the regime of β, we find three different behaviors for the limiting fluctuations whose covariances are explicitly computed. In particular, for the critical value β=1, starting a tagged particle near the slow bond, we obtain a family of Gaussian processes indexed in α, interpolating a fractional Brownian motion of Hurst exponent 1/4 and the degenerate process equal to zero. 相似文献
15.
It is proved that the solutions to the singular stochastic p-Laplace equation, p∈(1,2) and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r∈(0,1) on a bounded open domain Λ⊂Rd with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p and r respectively (in the Hilbert spaces L2(Λ), H−1(Λ) respectively). The highly singular limit case p=1 is treated with the help of stochastic evolution variational inequalities, where P-a.s. convergence, uniformly in time, is established. 相似文献
16.
In this paper we establish the boundedness of the extremal solution u∗ in dimension N=4 of the semilinear elliptic equation −Δu=λf(u), in a general smooth bounded domain Ω⊂RN, with Dirichlet data u|∂Ω=0, where f is a C1 positive, nondecreasing and convex function in [0,∞) such that f(s)/s→∞ as s→∞. 相似文献
17.
Let x(s), s∈Rd be a Gaussian self-similar random process of index H. We consider the problem of log-asymptotics for the probability pT that x(s), x(0)=0 does not exceed a fixed level in a star-shaped expanding domain T⋅Δ as T→∞. We solve the problem of the existence of the limit, θ?lim(−logpT)/(logT)D, T→∞, for the fractional Brownian sheet x(s), s∈[0,T]2 when D=2, and we estimate θ for the integrated fractional Brownian motion when D=1. 相似文献
18.
We study aspects of the analytic foundations of integration and closely related problems for functions of infinitely many variables x1,x2,…∈D. The setting is based on a reproducing kernel k for functions on D, a family of non-negative weights γu, where u varies over all finite subsets of N, and a probability measure ρ on D. We consider the weighted superposition K=∑uγuku of finite tensor products ku of k. Under mild assumptions we show that K is a reproducing kernel on a properly chosen domain in the sequence space DN, and that the reproducing kernel Hilbert space H(K) is the orthogonal sum of the spaces H(γuku). Integration on H(K) can be defined in two ways, via a canonical representer or with respect to the product measure ρN on DN. We relate both approaches and provide sufficient conditions for the two approaches to coincide. 相似文献
19.
Paul-Emile Maing 《Nonlinear Analysis: Theory, Methods & Applications》2008,68(12):3913-3922
This paper is concerned with the Cauchy problem for the fast diffusion equation ut−Δum=αup1 in RN (N≥1), where m∈(0,1), p1>1 and α>0. The initial condition u0 is assumed to be continuous, nonnegative and bounded. Using a technique of subsolutions, we set up sufficient conditions on the initial value u0 so that u(t,x) blows up in finite time, and we show how to get estimates on the profile of u(t,x) for small enough values of t>0. 相似文献
20.
In this paper, we consider the problem (Pε) : Δ2u=un+4/n-4+εu,u>0 in Ω,u=Δu=0 on ∂Ω, where Ω is a bounded and smooth domain in Rn,n>8 and ε>0. We analyze the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev inequality as ε→0 and we prove existence of solutions to (Pε) which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε small, (Pε) has at least as many solutions as the Ljusternik–Schnirelman category of Ω. 相似文献