The existence and nonexistence for time nonlocal evolution equations with superlinear sources

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.