AI 中文总结
研究具有超线性源的时间非局部发展方程初值问题，通过从属和单调迭代方法，分析其可解行为，建立存在性与不存在性充分条件，结果与初值指标相关且有临界维数现象，还通过例子展示适用性。
AI 中文摘要
我们考虑具有$(\mathcal{PC})$型核的时间非局部发展方程初值问题的可解行为。目的是分析一些充分条件，确保在非线性项快速增长时局部解的存在性和温和解的可积性。还建立了不可解结果的充分条件。可解行为与初值上的指标密切相关，出现临界维数现象。证明依赖于从属和单调迭代方法。最后给出几个例子说明结果的广泛适用性。
英文摘要
We consider the solvable behaviour for the initial value problem of time nonlocal evolution equations with the kernels of type $(\mathcal{PC})$. Our aim is to analyze some sufficient conditions ensuring local existence, integrability of mild solutions when the nonlinear term exhibits rapid growth. Moreover, a sufficient condition of the unsolvable result is also established. It turns out that the solvable behaviour is closely connected with the index on the initial value, which occurs a critical dimension phenomenon. The proofs rely on subordination and monotone iterative method. Finally, several examples are given to illustrate the wide applicability of the results.
