On nonlinear Sobolev equation with the Caputo fractional operator and exponential nonlinearity

The initial value problem for the Caputo type time‐fractional Sobolev equation with a nonlinear exponential source function is investigated in this work. We establish the existence and uniqueness of mild solutions corresponding to two different initial data assumptions. We derive global results of a unique mild solution with small initial data using some Sobolev/Sobolev‐Orlicz embeddings, a weighted Banach space, and the fixed point theorem. In the absence of any smallness assumptions, the Cauchy iteration method demonstrates that the mild solution blows up at a finite time or exists globally in time. Finally, we consider some illustrated examples to test the results obtained in theory.