Explosive solutions of stochastic nonlinear beam equations with damping

This study considers a class of damped stochastic nonlinear beam equations driven by multiplicative noise. By an appropriate energy inequality, we provide sufficient conditions such that the local solutions of the stochastic equations blow up with a positive probability or are explosive in an L²$L^2$ sense. We also derive estimates of the upper bound of the blow-up time.     

Quasi-static Maxwell's equations with a dissipative non-linear boundary condition: Full discretization

We study a time dependent eddy current equation for the magnetic field H accompanied with a nonlinear boundary condition, which is a generalization of the classical Silver–Müller condition for a non-perfect conductor. More exactly, the relation between the normal components of electric (E) and magnetic (H  ) fields obeys the following power law (linearized for small and large values) ν×E=ν×(|H×ν|^α−1H×ν)$\mathit{\nu }\times \mathit{E}=\mathit{\nu }\times ({|\mathit{H}\times \mathit{\nu }|}^{\alpha -1}\mathit{H}\times \mathit{\nu })$ for some α∈(0,1]$\alpha \in (0,1]$. We design a linear fully discrete approximation scheme to solve this nonlinear problem. The convergence of the approximations to a weak solution is proved, error estimates describing the dependence of the error on discretization parameters are derived as well. The efficiency of the proposed method is supported by numerical experiments.     

A sufficient condition for the existence of a principal eigenvalue for nonlocal diffusion equations with applications

Considerable work has gone into studying the properties of nonlocal diffusion equations. The existence of a principal eigenvalue has been a significant portion of this work. While there are good results for the existence of a principal eigenvalue equations on a bounded domain, few results exist for unbounded domains. On bounded domains, the Krein–Rutman theorem on Banach spaces is a common tool for showing existence. This article shows that generalized Krein–Rutman can be used on unbounded domains and that the theory of positive operators can serve as a powerful tool in the analysis of nonlocal diffusion equations. In particular, a useful sufficient condition for the existence of a principal eigenvalue is given.     

Convolution powers of complex functions on

Repeated convolution of a probability measure on leads to the central limit theorem and other limit theorems. This paper investigates what kinds of results remain without positivity. It reviews theorems due to Schoenberg, Greville, and Thomée which are motivated by applications to data smoothing (Schoenberg and Greville) and finite difference schemes (Thomée). Using Fourier transform arguments, we prove detailed decay bounds for convolution powers of finitely supported complex functions on . If M is an hermitian contraction, an estimate for the off‐diagonal entries of the powers of is obtained. This generalizes the Carne–Varopoulos Markov chain estimate.     

The Dirichlet problem for non‐divergence parabolic equations with discontinuous in time coefficients in a wedge

We consider the Dirichlet problem in a wedge for parabolic equation whose coefficients are measurable function of t. We obtain coercive estimates in weighted ‐spaces. The concept of “critical exponent” introduced in the paper plays here the crucial role. Various important properties of the critical exponent are proved. We give applications to the Dirichlet problem for linear and quasi‐linear non‐divergence parabolic equations with discontinuous in time coefficients in cylinders , where Ω is a bounded domain with an edge or with a conical point.     

Oscillation criteria for higher order nonlinear dynamic equations

In this paper, we will consider the higher‐order functional dynamic equations of the form on an above‐unbounded time scale , where and , . The function is a rd‐continuous function such that . The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.     

Long time behavior for nonlocal stochastic Kuramoto–Sivashinsky equations

In this paper, we consider a class of nonlocal stochastic Kuramoto–Sivashinsky equations driven by additive noises. Under some appropriate conditions, we investigate long time behavior, i.e., stability and growth bounds of the solutions to the equation. Finally, several examples are given to illustrate our results.