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

Abstract

It is well-known that in dimensions at least three semilinear wave equations with null conditions admit global solutions for small initial data. It is also known that in 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 dimension, waves do not decay. Nevertheless, we will prove that small data still lead to global solutions if the null condition is satisfied.

Global Existence and Long Time Behavior in the 1+1 dimensional Principal Chiral Model with Applications to Solitons

“…In one space dimension waves do not decay, and nonlinear resonance can lead to finite time blow up. Nevertheless, Luli, Yang and Yu in [30] proved, for Cauchy problems of semilinear wave equations with null conditions in one space dimension, the global existence of classical solutions with small initial data. The authors proposed a weighted energy and use the bootstrap method for obtain the result.…”

“…However, the factors Q 0 pφ, φq and Q 0 pφ, Λq provide decay and with the appropriate condition on λ, global regularity can be obtained. Inspired by Luli, Yang and Yu's result [30] in the semilinear case, it is natural to conjecture that the Cauchy problem for one-dimension system of quasilinear wave equations (3.1) admits a global classical solution for small initial data. The main aim of this theorem is to verify this conjecture.…”

“…The space time region D t0 is foliated by C t0,u 0 for u P R, and by C t0,u0 for u P R. Let us also consider the following energy estimate proposed in [4,30] for the scalar linear equations ψ " ρ given by: ż…”

“…Motivated by the above estimation (3.8) and [30], we define the space-time weighted energy norms valid for k " 0, 1:…”

“…Then, from this last condition and (3. Using the assumptions (3.11) and (3.12) the following pointwise bounds were established in [30].…”