Explosive solutions of a stochastic non‐local reaction–diffusion equation arising in shear band formation

Abstract

In this paper, we consider a non‐local stochastic parabolic equation that actually serves as a mathematical model describing the adiabatic shear banding formation phenomena in strained metals. We first present the derivation of the mathematical model. Then we investigate under which circumstances a finite‐time explosion for this non‐local stochastic partial differential equation, corresponding to shear banding formation, occurs. For that purpose, some results related to the maximum principle for this non‐local stochastic partial differential equation are derived, and afterwards the Kaplan eigenfunction method is employed. Copyright © 2015 John Wiley & Sons, Ltd.

A stochastic parabolic model of MEMS driven by fractional Brownian motion

“…It is also worth emphasizing the differences between the results in this paper for actuators driven by positively correlated noise (i.e., when H > 1/2) with existing literature focusing on the dynamics of the same systems when they are driven by standard Brownian motion; for more information on standard Brownian motion see Friedman (2006), Karatzas and Shreve (1991). It is of interest that Theorem 3 fails to provide an estimate of the quenching probability in the range (λ 1 , λ 1 + H 3 ).…”

Non-autonomous nonlocal partial differential equations with delay and memory

“…However, in recent decades, nonlocal (reaction-diffusion) problems have been investigated with great interest due to its usefulness in real applications (e.g. [1,3,16,20,26]).…”

“…One typical nonlocal problem, motivated by the mathematical modeling of a variety of phenomena coming from industrial applications ( [27]), and shear banding formation in high strain metals ( [3]), etc., has been studied by N. I. Kavallaris et al [20]. The authors considered the following nonlocal stochastic parabolic problem…”

Finite-time blow-up of a non-local stochastic parabolic problem

“…Here ξ is a F 0 -random variable in some suitable Hilbert spaces introduced later. The motivation of studying problem (1.1) -(1.3) is that this kind of non-local stochastic problems arise in the mathematical modelling of a variety of phenomena coming from industrial applications (e.g Ohmic heating in food sterilization [20,21,28] and shear banding formation in high strain metals [2,3,15]), biology (e.g. chemotaxis phenomenon [18,30]), statistical mechanics [19] and so on.…”

“…In particular, it represents the existence of external perturbations or a lack of knowledge of certain physical parameters which is actually quite often the case for this kind of systems. For a detailed construction of a mathematical model of the form (1.1) - (1.3) arising in shear banding formation in metals see [15].…”