论文标题
在较高力矩空间中的生长裂缝分离的光谱间隙上
On spectral gaps of growth-fragmentation semigroups in higher moment spaces
论文作者
论文摘要
我们提出了一种普遍的方法，以证明在瞬间空间中生长裂缝的半群的光谱差距和异步指数增长的存在$ l^{1}（\ Mathbb {r} _ {+} _ {+} _ {+}; \ x^{α} {α} dx） 1+x\right) ^{α}dx)$ for unbounded total fragmentation rates and continuous growth rates $r(.)$\ such that $\int_{0}^{+\infty } \frac{1}{r(τ)}dτ=+\infty .\ $The analysis is based on weak compactness tools and Frobenius theory of positive operators and holds provided that $α> \ wideHat {α} $对于合适的阈值$ \ wideHat {α} \ geq 1 $，取决于我们考虑的时刻空间。提供了系统的功能分析结构。给出了说明该理论的碎片核的各种例子，并提到了一个开放的问题。
We present a general approach to proving the existence of spectral gaps and asynchronous exponential growth for growth-fragmentation semigroups in moment spaces $L^{1}(\mathbb{R}_{+};\ x^{α}dx)$ and $L^{1}(\mathbb{R} _{+};\ \left( 1+x\right) ^{α}dx)$ for unbounded total fragmentation rates and continuous growth rates $r(.)$\ such that $\int_{0}^{+\infty } \frac{1}{r(τ)}dτ=+\infty .\ $The analysis is based on weak compactness tools and Frobenius theory of positive operators and holds provided that $α>\widehat{α}$ for a suitable threshold $\widehat{ α}\geq 1$ that depends on the moment space we consider. A systematic functional analytic construction is provided. Various examples of fragmentation kernels illustrating the theory are given and an open problem is mentioned.
