Long-time asymptotics for nonlinear growth-fragmentation equations
Long-time asymptotics for nonlinear growth-fragmentation equations
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DOI：
10.4310/cms.2012.v10.n3.a4
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发表时间：
2011-02
影响因子：
1
通讯作者：
Pierre Gabriel
中科院分区：
文献类型：
--
作者：
Pierre Gabriel
本文研究了一类带非线性增长项的增长-碎裂方程的长时间渐近行为。我们提出的例子，我们可以证明收敛到一个稳定的状态或相反的周期解的存在性。由于广义相对熵方法适用于精心选择的自相似解决方案，我们表明，该方程可以“渐近”减少到一个系统的常微分方程。然后利用李雅普诺夫泛函证明了系统的稳定性，并利用Poincare-Bendixon定理或Hopf分支证明了系统周期解的存在性.
We are interested in the long-time asymptotic behavior of growth-fragmentation equations with a nonlinear growth term. We present examples for which we can prove either the convergence to a steady state or conversely the existence of periodic solutions. Thanks the General Relative Entropy method applied to well chosen self-similar solutions, we show that the equation can ''asymptotically'' be reduced to a system of ODEs. Then stability results are proved by using a Lyapunov functional, and existence of periodic solutions are proved thanks to the Poincare-Bendixon theorem or by Hopf bifurcation.