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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
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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
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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
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A class of second-order abstract dissipative evolution differential operators D with 0∈kerD is shown for which the fact that a non-zero t?u(t) belongs to a cone and −Du to a dual cone may hold only on time intervals whose length is less than or equal to a defined number. Then oscillatory functions are dealt with in the framework of Banach spaces with a cone and conditions for the existence of a uniform oscillatory time for solutions of the equation Du=0 are given. 相似文献
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For α∈R, let pR(t,x,x) denote the diagonal of the transition density of the α-Bessel process in (0,1], killed at 0 and reflected at 1. As a function of x, if either α≥3 or α=1, then for t>0, the diagonal is nondecreasing. This monotonicity property fails if 1≠α<3. 相似文献
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We consider two-dimensional mixed problems in an exterior domain for a semilinear strongly damped wave equation with a power-type nonlinearity |u|p. If the initial data have a small weighted energy, we shall derive a global existence and energy decay results in the case when the power p of the nonlinear term satisfies p>6. 相似文献
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It is well known that the solution of the classical linear wave equation with an initial condition with compact support and vanishing initial velocity also has a compact support included in a set depending on time: the support of the solution at time t is causally related to that of the initial condition. Reed and Simon have shown that for a real-valued Klein–Gordon equation with (nonlinear) right-hand side −λu3 (λ>0), causality still holds. We show the same property for a one-dimensional Klein–Gordon problem but with transmission and with a more general repulsive nonlinear right-hand side F(u). We also prove the global existence of a solution using the repulsiveness of F. In the particular case F(u)=−λu3, the problem is a relativistic model for a quantum particle with repulsive self-interaction and tunnel effect at a semi-infinite potential step. 相似文献
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Razvan Gabriel Iagar Juan Luis Vázquez 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2009
We consider the Dirichlet problem for the p -Laplacian evolution equation, ut=Δpu, where p>2, posed in an exterior domain in RN, with zero Dirichlet boundary condition and with integrable and nonnegative initial data. We are interested in describing the influence of the holes of the domain on the large time behaviour of the solutions. Such behaviour varies depending on the relative values of N and p . We must distinguish between the behaviour near infinity of space (outer analysis), and near the holes (inner analysis). We obtain that the outer analysis is given in all cases by certain self-similar solutions and the inner analysis is given by quasi-stationary states. Logarithmic corrections to exact self-similarity appear in the critical case N=p, which is mathematically more interesting. In this first paper we treat only the cases N>p and N=p, the case N<p will be considered in a companion work. 相似文献
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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
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We study a family of differential operators Lα in two variables, depending on the coupling parameter α?0 that appears only in the boundary conditions. Our main concern is the spectral properties of Lα, which turn out to be quite different for α<1 and for α>1. In particular, Lα has a unique self-adjoint realization for α<1 and many such realizations for α>1. In the more difficult case α>1 an analysis of non-elliptic pseudodifferential operators in dimension one is involved. 相似文献
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In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given α>2, there are finitely many distance-regular graphs Γ with valency k, diameter D≥3 and v vertices satisfying v≤αk unless (D=3 and Γ is imprimitive) or (D=4 and Γ is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k≥3, diameter D≥3 and c2≥εk for a given 0<ε<1 unless (D=3 and Γ is imprimitive) or (D=4 and Γ is antipodal and bipartite). 相似文献