Weak Dispersive Estimates for Schrödinger Equations with Long Range Potentials

We prove local smoothing estimates for the Schrödinger initial value problem with data in the energy space L ²(? ^d ), d ≥ 2 and a general class of potentials. In the repulsive setting we have to assume just a power like decay (1 + |x|)^?γ for some γ > 0. Also attractive perturbations are considered. The estimates hold for all time and as a consequence a weak dispersion of the solution is obtained. The proofs are based on similar estimates for the corresponding stationary Helmholtz equation and Kato H-smooth theory.     

Modified Wave Operators Without Loss of Regularity for Some Long-Range Hartree Equations: I

We reconsider the theory of scattering for some long-range Hartree equations with potential |x|^?γ with 1/2 <  γ <  1. More precisely we study the local Cauchy problem with infinite initial time, which is the main step in the construction of the modified wave operators. We solve that problem in the whole subcritical range without loss of regularity between the asymptotic state and the solution, thereby recovering a result of Nakanishi. Our method starts from a different parametrization of the solutions, already used in our previous papers. This reduces the proofs to energy estimates and avoids delicate phase estimates.     

Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation

We study the action of metaplectic operators on Wiener amalgam spaces, giving upper bounds for their norms. As an application, we obtain new fixed-time estimates in these spaces for Schrödinger equations with general quadratic Hamiltonians and Strichartz estimates for the Schrödinger equation with potentials V(x)=±²|x|.     

Exponentially fitted one-step methods for the numerical solution of the scalar Riccati equation

Using stability analysis and information from the constant coefficient problem, we motivate an explicit exponentially fitted one-step method to approximate the solution of a scalar Riccati equation ϵy′ = c(x)y² + d(x)y + e(x), 0 < x ⩽ x, y(0) = y₀, where ϵ > 0 is a small parameter and the coefficients c, d and e are assumed to be real valued and continuous. An explicit Euler-type scheme is presented which, when applied to the numerical integration of the continuous problem, give solutions satisfying a uniform (in ϵ) error estimate with order one (where suitable restrictions are imposed on the coefficients c, d and e together with the choice of y(0)). Using a counterexample, we show that, for a particular class of problems, the solutions of the fitted scheme do not converge uniformly (in ϵ) to the corresponding solutions of the continuous problems. Numerical results are presented which compare the fitted scheme with a number of implicit schemes when applied to the numerical integration of some sample problems.     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

Local energy decay for the damped wave equation

We prove local energy decay for the damped wave equation on R^d${\mathbb{R}}^d$. The problem which we consider is given by a long range metric perturbation of the Euclidean Laplacian with a short range absorption index. Under a geometric control assumption on the dissipation we obtain an almost optimal polynomial decay for the energy in suitable weighted spaces. The proof relies on uniform estimates for the corresponding “resolvent”, both for low and high frequencies. These estimates are given by an improved dissipative version of Mourre's commutators method.     

The Semilinear Wave Equation on Asymptotically Euclidean Manifolds

We consider the quadratically semilinear wave equation on (? ^d , 𝔤), d ≥ 3. The metric 𝔤 is non-trapping and approaches the Euclidean metric like ?x?^?ρ. Using Mourre estimates and the Kato theory of smoothness, we obtain, for ρ > 0, a Keel–Smith–Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence for the nonlinear problem with small initial data for ρ ≥ 1. Long time existence means that, for all n > 0, the life time of the solution is a least δ^?n , where δ is the size of the initial data in some appropriate Sobolev space. Moreover, for d ≥ 4 and ρ > 1, we obtain global existence for small data.     

On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation

In this paper, we consider the Cauchy problem for the Helmholtz equation in a rectangle, where the Cauchy data is given for y=0 and boundary data are for x=0 and x=π. The solution is sought in the interval 0<y≤1. A quasi-reversibility method is applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

On a nonlinear eigenvalue problem in ODE

In this paper we shall study the following variant of the logistic equation with diffusion:

−du^″(x)=g(x)u(x)−u²(x)     

Decay estimates for nonlinear nonlocal diffusion problems in the whole space

In this paper, we obtain bounds for the decay rate in the L ^r (? ^d )-norm for the solutions of a nonlocal and nonlinear evolution equation, namely, $$u_t \left( {x,t} \right) = \int_{\mathbb{R}^d } {K\left( {x,y} \right)\left| {u\left( {y,t} \right) - u\left( {x,t} \right)} \right|^{p - 2} \left( {u\left( {y,t} \right) - u\left( {x,t} \right)} \right)dy, x \in \mathbb{R}^d , t > 0.}$$ . We consider a kernel of the form K(x, y) = ψ(y?a(x)) + ψ(x?a(y)), where ψ is a bounded, nonnegative function supported in the unit ball and a is a linear function a(x) = Ax. To obtain the decay rates, we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form $$T\left( u \right) = - \int_{\mathbb{R}^d } {K\left( {x,y} \right)\left| {u\left( y \right) - u\left( x \right)} \right|^{p - 2} \left( {u\left( y \right) - u\left( x \right)} \right)dy, 1 \leqslant p < \infty .}$$ . The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole space ? ^d : $$\lambda _{1,p} \left( {\mathbb{R}^d } \right) = 2\left( {\int_{\mathbb{R}^d } {\psi \left( z \right)dz} } \right)\left| {\frac{1} {{\left| {\det A} \right|^{1/p} }} - 1} \right|^p .$$ Moreover, we deal with the p = ∞ eigenvalue problem, studying the limit of λ _1,p ^1/p as p→∞.     

Existence and Concentration of Bound States of a Class of Nonlinear SchrSdinger Equations in R2 with Potential Tending to Zero at Infinity

In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schr?dinger equation $$- \varepsilon ^2 \Delta u_\varepsilon + V\left( x \right)u_\varepsilon = K\left( x \right)\left| {u_\varepsilon } \right|^{p - 2} u_\varepsilon e^{\alpha _0 \left| {u_\varepsilon } \right|^\gamma } , u_\varepsilon > 0, u_\varepsilon \in H^1 \left( {\mathbb{R}^2 } \right),$$ where 2 < p < ∞, α ₀ > 0, 0 < γ < 2. When the potential function V (x) decays at infinity like (1 + |x|)^?α with 0 < α ≤ 2 and K(x) > 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H ¹(?²)-solution u _? exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schr?dinger equation $- \Delta u + V\left( \xi \right)u = K\left( \xi \right)\left| u \right|^{p - 2} ue^{\alpha _0 \left| u \right|^\gamma }$ has local minimum points. Furthermore, the concentration property of u _? is also established as ? tends to zero.     

About the p-adic Yosida equation inside a disk

Let K be a complete ultrametric algebraically closed field and let ?(d(0, R^?)) be the field of meromorphic functions inside the disk d(0,R⁻) = {x ∈ K ∣ ∣x∣ < R}. Let ?_b(d(0, R^?)) be the subfield of bounded meromorphic functions inside d(0,R⁻) and let ?_u(d(0, R^?)) = ?(d(0, R^?)) ? ?_b(d(0, R^?)) be the subset of unbounded meromorphic functions inside d(0,R⁻). Initially, we consider the Yosida Equation: , where m ∈ ?^* and F(X) is a rational function of degree d with coefficients in ?_b(d(0, R^?)). We show that, if d ≥ 2m + 1, this equation has no solution in ?_u(d(0, R^?)).Next, we examine solutions of the above equation when F(X) is apolynomial with constant coefficients and show that it has no unbounded analytic functions in d(0,R⁻). Further, we list the only cases when the equation may eventually admit solutions in ?_u(d(0, R^?)). Particularly, the elliptic equation may not.     

Concerning the wave equation on asymptotically Euclidean manifolds

We obtain KSS, Strichartz and certain weighted Strichartz estimates for the wave equation on (ℝ ^d , g), d ≥ 3, when the metric g is non-trapping and approaches the Euclidean metric like 〈x〉^−ρ with ρ > 0. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for ρ > 1 and d = 3. Also, we establish the Strauss conjecture when the metric is radial with ρ > 1 for d = 3.     

Riesz extremal measures on the sphere for axis-supported external fields

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere Sd in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x−y|^−s with d−2?s<d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on Sd is determined. The special case s=d−2 yields interesting phenomena, which we investigate in detail. A weak^∗ asymptotic analysis is provided as s→⁺(d−2).     

Hausdorff dimension of Lebesgue sets for W α p classes on metric spaces

Let (X, μ, d) be a space of homogeneous type, where d and p are a metric and a measure, respectively, related to each other by the doubling condition with γ > 0. Let W _α ^p (X) be generalized Sobolev classes, let Cap_{α p} (where p > 1 and 0 < α ≤ 1) be the corresponding capacity, and let dim_H be the Hausdorff dimension. We show that the capacity Cap_{α p} is related to the Hausdorff dimension; we also prove that, for each function u ∈ W _α/p (X), p > 1, 0 < a < γ/p, there exists a set E ? X such that dim _H (E) ≤ γ - αp, the limit $$\mathop {\lim }\limits_{r \to + 0} \frac{1}{{\mu (B(x,r))}}\int_{B(x,r)} {u d\mu = u * (x)} $$ exists for each x ∈ X\E, and moreover $$\mathop {\lim }\limits_{r \to + 0} \frac{1}{{\mu (B(x,r))}}\int_{B(x,r)} {\left| {u - u * (x)} \right|^q d\mu = 0, \frac{1}{q} = \frac{1}{p} - \frac{\alpha }{\gamma }.} $$ .     

Existence of positive solutions of nonlinear fractional differential equations

Existence of positive solutions for the nonlinear fractional differential equation D^su(x)=f(x,u(x)), 0<s<1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804-812), where D^s denotes Riemann-Liouville fractional derivative. In the present work we study existence of positive solutions in case of the nonlinear fractional differential equation:

L(D)u=f(x,u),u(0)=0,0<x<1,     

Multiple positive solutions to a singular boundary value problem for a superlinear Emden-Fowler equation

A multiplicity result for the singular ordinary differential equation y^″+λx⁻²yσ=0, posed in the interval (0,1), with the boundary conditions y(0)=0 and y(1)=γ, where σ>1, λ>0 and γ?0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ^?∈(0,σ/2] such that a solution to the above problem is possible if and only if λγσ−1?Σ^?. For 0<λγσ−1<Σ^?, there are multiple positive solutions, while if γ=(λ⁻¹Σ^?)^1/(σ−1) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y(x) as x→⁰⁺ is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem −Δu=d⁻²(x)uσ in Ω, where Ω⊂RN, N?2, is a smooth bounded domain and d(x)=dist(x,∂Ω).     

Determining surface heat flux in the steady state for the Cauchy problem for the Laplace equation

In this paper, we consider the Cauchy problem for the Laplace equation, in a strip where the Cauchy data is given at x = 0 and the flux is sought in the interval 0<x?1. This problem is typical ill-posed: the solution (if it exists) does not depend continuously on the data. We study a modification of the equation, where a fourth-order mixed derivative term is added. Some error stability estimates for the flux are given, which show that the solution of the modified equation is approximate to the solution of the Cauchy problem for the Laplace equation. Furthermore, numerical examples show that the modified method works effectively.     

Uniform boundary Harnack principle and generalized triangle property

Let D be a bounded open subset in R^d, d?2, and let G denote the Green function for D with respect to (-Δ)^α/2, 0<α?2, α<d. If α<2, assume that D satisfies the interior corkscrew condition; if α=2, i.e., if G is the classical Green function on D, assume—more restrictively—that D is a uniform domain. Let g=G(·,y₀)∧1 for some y₀∈D. Based on the uniform boundary Harnack principle, it is shown that G has the generalized triangle property which states that when d(z,x)?d(z,y). An intermediate step is the approximation G(x,y)≈|x-y|^α-dg(x)g(y)/g(A)², where A is an arbitrary point in a certain set B(x,y).This is discussed in a general setting where D is a dense open subset of a compact metric space satisfying the interior corkscrew condition and G is a quasi-symmetric positive numerical function on D×D which has locally polynomial decay and satisfies Harnack's inequality. Under these assumptions, the uniform boundary Harnack principle, the approximation for G, and the generalized triangle property turn out to be equivalent.     

On two-sided and asymptotic estimates for the norms of embedding operators of into L q (dμ)

Explicit upper and lower estimates are given for the norms of the operators of embedding of , n ∈ ?, in L _q (dµ), 0 < q < ∞. Conditions on the measure µ are obtained under which the ratio of the above estimates tends to 1 as n → ∞, and asymptotic formulas are presented for these norms in regular cases. As a corollary, an asymptotic formula (as n → ∞) is established for the minimum eigenvalues λ_{1, n, β} , β > 0, of the boundary value problems (?d ²/dx ²) ⁿ u(x) = λ|x| ^β?1, x ∈ (?1, 1), u ^(k)(±1) = 0, k ∈ {0, 1, ..., n ? 1}.