On one-dimension semi-linear wave equations with null conditions

It is well-known that in space-dimensions at least three semilinear wave equations with null conditions admit global solutions for small initial data. It is also known that in space-dimension two such result still holds for a certain class of quasi-linear wave equations with null conditions. The proofs are based on the decay mechanism of linear waves. However, in one space-dimension, waves do not decay. Nevertheless, we will prove that small data still lead to global solutions if the null condition is satisfied.