Random Hermite differential equations: Mean square power series solutions and statistical properties

This paper deals with the construction of random power series solution of second order linear differential equations of Hermite containing uncertainty through its coefficients and initial conditions. Under appropriate hypotheses on the data, we establish that the constructed random power series solution is mean square convergent. We provide conditions in order to obtain random polynomial solutions and, as a consequence, random Hermite polynomial are introduced. Also, the main statistical functions of the approximate stochastic process solution generated by truncation of the exact power series solution are given. Finally, we apply the proposed technique to several illustrative examples comparing the numerical results with respect to those provided by other available approaches including Monte Carlo simulation.     

A class of evolution equations: Existence of solutions with functional boundary conditions

We consider the evolution equation whose right-hand side is the sum of a linear unbounded operator generating a compact strongly continuous semigroup and a continuous operator acting in function spaces. We prove the existence of a solution that stays within a given closed convex set and moreover, satisfies a functional boundary condition, particular cases of which are the Cauchy initial condition, periodicity condition, mixed condition including continuous transformations of spatial variables, etc. The main result is illustrated by using an example of the boundary-value problem for a partial operator-differential equation. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 48–60, January, 1999.     

Self-similar solutions of a nonlinear friction equation in higher dimensions

We study the class of self-similar solutions of certain multi-dimensional kinetic models of granular flows, which have been recently introduced in connection with the quasi elastic limit of a model Boltzmann equation with dissipative collisions and variable coefficient of restitution. The importance of these solutions in connection with the cooling of the dissipative gas is subsequently discussed.

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Sunto Si studia la classe delle soluzioni di similarità di alcune equazioni cinetiche per flussi granulari in più dimensioni. Queste equazioni sono state introdotte di recente in connessione con il limite quasi elastico di un’ equazione di Boltzmann per collisioni dissipative con coefficiente di restituzione variabile. Nella seconda parte del lavoro si discute l’importanza di tali soluzioni nello studio del raffreddamento del gas dissipativo.

     

Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations

In this paper we consider the semilinear elliptic problem Δu=a(x)f(u), u?0 in Ω, with the boundary blow-up condition u_|∂Ω=+∞, where Ω is a bounded domain in RN(N?2), a(x)∈C(Ω) may blow up on ∂Ω and f is assumed to satisfy (f₁) and (f₂) below which include the sublinear case f(u)=um, m∈(0,1). For the radial case that Ω=B (the unit ball) and a(x) is radial, we show that a solution exists if and only if . For Ω a general domain, we obtain an optimal nonexistence result. The existence for nonradial solutions is also studied by using sub-supersolution method.     

含积分边界条件的分数阶微分方程边值问题的正解的存在性

研究了含积分边界条件的分数阶微分方程的边值问题,首先给出格林函数及性质,其次将问题转化为一个等价的积分方程,最后应用Krasnoselkii及Leggett-Williams不动点定理得到了一个及多个正解的存在性,推广了以往的结果．     

The quasiderivative method for derivative estimates of solutions to degenerate elliptic equations

We give an example of quasiderivatives constructed by random time change, Girsanov’s Theorem and Levy’s Theorem. As an application, we investigate the smoothness and estimate the derivatives up to second order for the probabilistic solution to the Dirichlet problem for the linear degenerate elliptic partial differential equation of second order, under the assumption of non-degeneracy with respect to the normal to the boundary and an interior condition to control the moments of quasiderivatives, which is weaker than non-degeneracy.     

A kinetic approach to comparison properties for degenerate parabolic-hyperbolic equations with boundary conditions

We study the comparison principle for degenerate parabolic-hyperbolic equations with initial and nonhomogeneous boundary conditions. We prove a comparison theorem for any entropy sub- and supersolution. The L¹ contractivity and, therefore, uniqueness of entropy solutions has been obtained so far by some authors, but it seems that any comparison theorem is not proven. The method used there is the doubling variable technique due to Kru?kov. Our method is based upon the kinetic formulation and the kinetic techniques. By developing the kinetic techniques for degenerate parabolic-hyperbolic equations with boundary conditions, we can obtain a comparison property which obviously extends the L¹ contractive property.     

Nontrivial solutions of boundary value problems of second-order difference equations

We employ the critical point theory to establish the existence of nontrivial solutions for some boundary value problems of second-order difference equations.     

A hyperbolicity criterion for periodic solutions of functional-differential equations: The case of rational periods

We establish necessary and sufficient conditions for hyperbolicity of periodic solutions of nonlinear functional-differential equations.     

Self-similar solutions of quasilinear parabolic equations with nonlinear gradient terms

This paper is devoted to study the classification of self-similar solutions to the m ≥ 1,p,q ＞ 0 and p + q ＞ m. For m = 1, it is shown that the very singular self-similar solution exists if and only if nq + (n + 1)p ＜ n + 2, and in case of existence, such solution is unique. For m ＞ 1, it is shown that very singular self-similar solutions exist if and only if 1 ＜ m ＜ 2 and nq + (n + 1)p ＜ 2 + mn, and such solutions have compact support if they exist. Moreover, the interface relation is obtained.     

Two positive solutions of a boundary value problem for difference equations

We consider a second order vector boundary value problem for difference equations and establish criteria for the existence of at least two positive solutions by an application of a fixed point theorem in cones.     

Blow-up of solutions to semilinear wave equations with variable coefficients and boundary

This paper is devoted to studying initial-boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions. We will establish blow-up results for the initial-boundary value problems. It is proved that there can be no global solutions no matter how small the initial data are, and also we give the life span estimate of solutions for the problems.     

On solutions of stochastic differential equations with parameters modeled by random sets

We consider ordinary stochastic differential equations whose coefficients depend on parameters. After giving conditions under which the solution processes continuously depend on the parameters random compact sets are used to model the parameter uncertainty. This leads to continuous set-valued stochastic processes whose properties are investigated. Furthermore, we define analogues of first entrance times for set-valued processes called first entrance and inclusion times. The theoretical concept is applied to a simple example from mechanics.