The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted \(L^1\) spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the solutions to the Cauchy problem, resulting from the unboundedness of the total fragmentation rate. It allows us to prove the quasi-compactness of the associated (rescaled) semigroup, which in turn provides the exponential convergence toward the projector on the Perron eigenfunction.

中文翻译：

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具有无穷碎片率的生长碎片方程的异步指数增长

目的是证明在大的加权\（L ^ 1 \）空间中以及在一般系数假设下，增长碎片方程的异步指数增长。关键的论据是柯西问题解决方案的动因，这是总碎片率无限制的结果。它使我们能够证明相关的（重新缩放的）半群的拟紧凑性，从而在Perron本征函数上向投影机提供指数收敛。