Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems

We study a continuous-time stochastic process on strings made of two types of particle, whose dynamics mimic the behaviour of microtubules in a living cell; namely, the strings evolve via a competition between (local) growth/shrinking as well as (global) hydrolysis processes. We give a complete characterization of the phase diagram of the model, and derive several criteria of the transient and recurrent regimes for the underlying stochastic process.

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Long-Time Behaviour in a Model of Microtubule Growth

Advances in Applied Probability ◽

10.1017/s0001867800004006 ◽

2010 ◽

Vol 42 (01) ◽

pp. 268-291 ◽

Cited By ~ 1

Author(s):

O. Hryniv ◽

M. Menshikov

Keyword(s):

Stochastic Process ◽

Continuous Time ◽

Complete Characterization ◽

Time Behaviour ◽

Local Growth ◽

Long Time Behaviour ◽

Long Time ◽

Microtubule Growth ◽

Continuous Time Stochastic Process

We study a continuous-time stochastic process on strings made of two types of particle, whose dynamics mimic the behaviour of microtubules in a living cell; namely, the strings evolve via a competition between (local) growth/shrinking as well as (global) hydrolysis processes. We give a complete characterization of the phase diagram of the model, and derive several criteria of the transient and recurrent regimes for the underlying stochastic process.

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Long-time behaviour of solutions to a one-dimensional strongly nonlinear model for phase transitions with micro-movements

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210511000229 ◽

2012 ◽

Vol 142 (6) ◽

pp. 1279-1307

Author(s):

Jie Jiang

Keyword(s):

Phase Transitions ◽

Nonlinear Partial Differential Equation ◽

Global Solutions ◽

Global Attractors ◽

Equation System ◽

Time Behaviour ◽

One Dimensional ◽

Strongly Nonlinear ◽

Long Time Behaviour ◽

Long Time

This paper studies the long-time behaviour of solutions to a one-dimensional strongly nonlinear partial differential equation system arising from phase transitions with microscopic movements. Our system features a strongly nonlinear internal energy balance equation. Uniform bounds of the global solutions and the compactness of the orbit are obtained for the first time using a lemma established recently by Jiang. The existence of global attractors and convergence of global solutions to a single steady state as time goes to infinity are also proved.

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