The Cauchy problem for the logarithmic Schr\"odinger equation revisited

TL;DR

This paper revisits the Cauchy problem for the logarithmic Schrödinger equation, constructing strong solutions in various energy spaces without relying on compactness, thus providing a constructive approach to solution existence.

Contribution

It introduces a new constructive method for solving the Cauchy problem in energy spaces for the logarithmic Schrödinger equation, avoiding traditional compactness arguments.

Findings

Constructed strong solutions in $H^1$ and $H^2$ energy spaces.

Provided a constructive approach avoiding compactness arguments.

Showed solutions form a Cauchy sequence leading to convergence.

Abstract

We revisit the Cauchy problem for the logarithmic Schr\"odinger equation and construct strong solutions in $H^1$, the energy space, and the $H^2$-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.