On the existence time of local solutions for critical semilinear Schrödinger equations in Sobolev spaces

The existence time of local solutions of semilinear Schrödinger equations in Sobolev spaces is considered based on the method of frequency decomposition. The semilinear terms are power type or exponential type, which are critical in terms of the scaling or the Trudinger–Moser inequality. The existence time is estimated by the low and high frequency parts of data.