# A sharp Lorentz-invariant Strichartz norm expansion for the cubic wave equation in $\mathbb{R}^{1+3}$

## TL;DR

This paper derives an asymptotic expansion for the maximal Strichartz norm of small solutions to the cubic wave equation in Minkowski space, highlighting the influence of focusing or defocusing nonlinearity on the second-order term.

## Contribution

It provides the first explicit calculation of the second-term constant in the Strichartz norm expansion for the cubic wave equation, distinguishing focusing and defocusing cases.

## Key findings

- Leading coefficient matches Foschi's sharp constant.
- Second-term constant differs for focusing vs. defocusing cases.
- Sign of the second-term coefficient changes with nonlinearity type.

## Abstract

We provide an asymptotic formula for the maximal Strichartz norm of small solutions to the cubic wave equation in Minkowski space. The leading coefficient is given by Foschi's sharp constant for the linear Strichartz estimate. We calculate the constant in the second term, which differs depending on whether the equation is focussing or defocussing. The sign of this coefficient also changes accordingly.