The Cauchy Problem for the Logarithmic Schrödinger Equation Revisited

We revisit the Cauchy problem for the logarithmic Schrödinger equation and construct strong solutions in \(H^1\) $H^1$ , the energy space, and the \(H^2\) $H^2$ -energy space. The solutions are provided in a constructive way, which does not rely on compactness arguments, that a sequence of approximate solutions forms a Cauchy sequence in a complete function space and then actual convergence is shown to be in a strong sense.