The existence and nonexistence for time nonlocal evolution equations with superlinear sources
arXiv:2607.13177
The paper studies initial value problems for time‑nonlocal evolution equations with PC kernels, providing conditions for local existence and integrability of mild solutions with superlinear sources, and also establishing criteria for nonexistence, highlighting a critical dimension phenomenon.
Abstract
We consider the solvable behaviour for the initial value problem of time nonlocal evolution equations with the kernels of type . 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.