On the principal eigenvalue of some nonlocal diffusion problems

In this paper we analyze some properties of the principal eigenvalue λ₁(Ω) of the nonlocal Dirichlet problem (J∗u)(x)−u(x)=−λu(x) in Ω with u(x)=0 in RN?Ω. Here Ω is a smooth bounded domain of RN and the kernel J is assumed to be a C¹ compactly supported, even, nonnegative function with unit integral. Among other properties, we show that λ₁(Ω) is continuous (or even differentiable) with respect to continuous (differentiable) perturbations of the domain Ω. We also provide an explicit formula for the derivative. Finally, we analyze the asymptotic behavior of the decreasing function Λ(γ)=λ₁(γΩ) when the dilatation parameter γ>0 tends to zero or to infinity.     

Study of the solutions to a model porous medium equation with variable exponent of nonlinearity

The authors of this paper study the Dirichlet problem of the following equation

ut−div(|u|^ν(x,t)∇u)=f−|u|^p(x,t)−1u.     

Boundary layer solutions to problems with infinite-dimensional singular and regular perturbations

We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed equations of the type (ε²(x)u^′′(x))=f(x,u(x))+g(x,u(x),ε(x)u^′(x)), 0<x<1, with Dirichlet and Neumann boundary conditions. Here the functions ε and g are small and, hence, regarded as singular and regular functional perturbation parameters. The main tool of the proofs is a generalization (to Banach space bundles) of an implicit function theorem of R. Magnus.     

The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function

The Nehari manifold for the equation −Δu(x)=λa(x)u(x)+b(x)|u(x)|^ν−1u(x) for x∈Ω together with Dirichlet boundary conditions is investigated. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form t→J(tu) where J is the Euler functional associated with the equation) we discuss how the Nehari manifold changes as λ changes and show how existence and non-existence results for positive solutions of the equation are linked to properties of the manifold.     

Integral representations of harmonic functions in half spaces

In this paper, using a modified Poisson kernel in an upper half-space, we prove that a harmonic function u(z) in a upper half space with its positive part u⁺(x)=max{u(x),0} satisfying a slowly growing condition can be represented by its integral in the boundary of the upper half space, the integral representation is unique up to the addition of a harmonic polynomial, vanishing in the boundary of the upper half space and that its negative part u⁻(x)=max{−u(x),0} can be dominated by a similar slowly growing condition, this improves some classical result about harmonic functions in the upper half space.     

Spectral properties of the neutron transport equation

The general equation describing the steady-state flow through a porous column is λu ? D_xA(D_x?(u) + G(u)) = f, where λ is a nonnegative constant. In this paper existence, uniqueness and comparison results for solutions to the Dirichlet and mixed boundary value problems associated with this equation are proven. The existence of a weak solution to the evolution problems associated with the equation u_t = D_x(D_x?(u) + G(u)) are deduced.     

A practical choice of parameters in improved SOR-Newton method with orderings

In this paper, on the basis of the results of Ishihara et al. (1997), we first discuss global convergence theorems for the improved SOR-Newton and block SOR-Newton methods with orderings applied to a system of mildly nonlinear equations, which includes as a special case the discretized version of the Dirichlet problem, for the equation ϵΔu + p(x)u_x + q(y)u_y = f(x, y, u), where f is continuously differentiable and fu(x, y, u) ⩾ 0. Moreover, we propose a practical choice of the multiple relaxation parameters {ω_i} for them. Numerical examples are also given.     

Weighted doubling properties and unique continuation theorems for the degenerate Schrödinger equations with singular potentials

Let u be the weak solution to the degenerate Schrödinger equation with singular coefficients in Lipschitz domain as following

−div(w(x)A(x)∇u(x))+V(x)u(x)w(x)=0,     

Periodic solutions for wave equations with variable coefficients with nonlinear localized damping

We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g^′(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain.     

Stability of the Absolutely Continuous Spectrum of Random Schrödinger Operators on Tree Graphs

According to the Smolukowski-Kramers approximation, we show that the solution of the semi-linear stochastic damped wave equations μ u _tt (t,x)=Δu(t,x)?u _t (t,x)+b(x,u(t,x))+Q (t),u(0)=u ₀, u _t (0)=v ₀, endowed with Dirichlet boundary conditions, converges as μ goes to zero to the solution of the semi-linear stochastic heat equation u _t (t,x)=Δ u(t,x)+b(x,u(t,x))+Q (t),u(0)=u ₀, endowed with Dirichlet boundary conditions. Moreover we consider relations between asymptotics for the heat and for the wave equation. More precisely we show that in the gradient case the invariant measure of the heat equation coincides with the stationary distributions of the wave equation, for any μ>0.     

A nonhomogeneous elliptic problem involving critical growth in dimension two

In this paper we study a class of nonhomogeneous Schrödinger equations

−Δu+V(x)u=f(u)+h(x)     

On the Morse critical groups for indefinite sublinear elliptic problems

We consider the Dirichlet problem for the equation −Δu=αu+m(x)u|u|^q−2+g(x,u), where q∈(1,2) and m changes sign. We prove that the Morse critical groups at zero of the energy functional of the problem are trivial. As a consequence, existence and bifurcation of nontrivial solutions of the problem are established.     

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)     

On inverse scattering for the wave equation with a potential term via the Lax-Phillips theory: A simple proof

The inverse scattering problem for the perturbed wave equation (1) □u + V(x)u = 0 in $\text{Ω=R}{}^{\mathrm{n}}$ (n = odd ? 3) is considered. Here the potentials V(x) are real, smooth, with compact support and non-negative. We apply the Lax and Phillips theory, together with some properties of solutions of a Dirichlet problem associated with the operator ?Δ + V(x) to show, in a very simple way, that the scattering operator S(V) associated with (1) determines uniquely the scatterer, provided that a fixed sign condition on the potentials is satisfied. We also show that the map V → S(V) is once-differentiable.     

Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

We consider the Dirichlet problem for the equation −Δu=λu±f(x,u)+h(x) in a bounded domain, where f has a sublinear growth and h∈L². We find suitable conditions on f and h in order to have at least two solutions for λ near to an eigenvalue of −Δ. A typical example to which our results apply is when f(x,u) behaves at infinity like a(x)|u|^q−2u, with M>a(x)>δ>0, and 1<q<2.     

Periodic solutions of the nonlinear telegraph equations with bounded nonlinearities

In this article, using the Leray-Schauder degree theory, we discuss existence, nonexistence and multiplicity for the periodic solutions of the nonlinear telegraph equation

utt−uxx+cut+Φ(u)=f(t,x)+s,     

A kernel for automorphic L-functions

The full multiple Dirichlet series of an automorphic cusp form is defined, in classical language, as a Dirichlet series of several complex variables over all the Fourier coefficients of the cusp form. It is different from the L-function of Godement and Jacquet, which is defined as a Dirichlet series in one complex variable over a one-dimensional array of the Fourier coefficients. In GL(2) and GL(3), the two notions are simply related. In this paper, we construct a kernel function that gives the full multiple Dirichlet series of automorphic cusp forms on GL(n,R). The kernel function is a new Poincaré series. Specifically, the inner product of a cusp form with this Poincaré series is the product of the full multiple Dirichlet series of the form times a function that is essentially the Mellin transform of Jacquet's Whittaker function. In the proof, the full multiple Dirichlet series is produced by applying the Lipschitz summation formula several times and by an integral which collapses the sum over SL(n−1,Z) in the Fourier expansion of the cusp form.     

Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach

In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation ut(x,t)=(k(x)uxx(x,t)), with Dirichlet boundary conditions u(0,t)=ψ₀, u(1,t)=ψ₁. Main goal of this study is to investigate the distinguishability of the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[⋅] and Ψ[⋅] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x=0 and x=1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f(t):=k(0)ux(0,t) or/and h(t):=k(1)ux(1,t). In addition to these, the values k^′(0) and k^′(1) of the unknown coefficient k(x) at x=0 and x=1, respectively, are also determined via the input data. Furthermore, it is shown that measured output dataf(t) and h(t) can be determined analytically, by an integral representation. Hence the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x=0 and x=1.     

Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay

In this paper, we study the existence, uniqueness and asymptotic stability of travelling wavefronts of the following equation:

u_t(x,t)=D[u(x+1,t)+u(x-1,t)-2u(x,t)]-du(x,t)+b(u(x,t-r)),     

Phragmen-Lindelöf decay theorems for classes of nonlinear Dirichlet problems in a circular cylinder

Classes of nonlinear elliptic equations in a long circular cylinder of radius one are considered. The equations are of the form ▽²u = S(u, u′)u″ + T(u)u′², where u = u(x₁, x₂, x₃), and u′, u″ represent general partial derivatives of the indicated order. Homogeneous Dirichlet data are prescribed on the long sides of the cylinder, and throughout the cylinder u is a priori assumed to be sufficiently small while u′ (and, for some classes, also u″) is assumed to be bounded in absolute value by one. With the above assumptions, it is proved that every solution u decays exponentially with distance from the nearer end with a decay constant k which depends on the smoothness properties of S and T but is independent of the length of the cylinder.