Blowing up solutions for cubic one dimensional Klein-Gordon equations
Abstract
Consider a quasi-linear (or semi-linear) cubic Klein-Gordon equation in one space dimension with small, smooth and decaying initial data. It is known that when the nonlinearity satisfies a convenient ``null condition'', the solution is global. It is conjectured that, if this null condition is not satisfied, generic initial data should give rise to solutions blowing up at a time of magnitude similar to ± eS*/ϵ2, where S*>0 and $0<\epsilon\ll 1 $ is the size of the initial data. We consider in this monograph equation (∂t2−∂x2 +1)u = (∂t u)3, where the cubic semi-linear nonlinearity does not satisfy the null condition. We construct a blowing up solution at time T*(ϵ) = eS*/ϵ2, which at time T0(ϵ) = ϵ−2T*(ϵ)1−b (b>0 small), satisfies smallness and decay conditions, compatible with those allowing one to prove global existence for the similar problem when the nonlinearity satisfies the null condition. Moreover, we have an asymptotic description of u close to the unique blowing up point (T*(ϵ),0).
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BibTeX
@misc{delort2026blowing,
title = {{Blowing up solutions for cubic one dimensional Klein-Gordon equations}},
author = {Jean-Marc Delort},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05751180v1},
note = {CC BY-NC 4.0},
}
RIS
TY - UNPB
TI - Blowing up solutions for cubic one dimensional Klein-Gordon equations
AU - Jean-Marc Delort
PY - 2026
DA - 2026/09/15
PB - HAL
UR - https://hal.science/hal-05751180v1
AB - Consider a quasi-linear (or semi-linear) cubic Klein-Gordon equation in one space dimension with small, smooth and decaying initial data. It is known that when the nonlinearity satisfies a convenient ``null condition'', the solution is global. It is conjectured that, if this null condition is not satisfied, generic initial data should give rise to solutions blowing up at a time of magnitude similar to ± e^S_*/ϵ^2, where S_*>0 and $0<\epsilon\ll 1 $ is the size of the initial data. We consider in this monograph equation (∂_t^2−∂_x^2 +1)u = (∂_t u)^3, where the cubic semi-linear nonlinearity does not satisfy the null condition. We construct a blowing up solution at time T_*(ϵ) = e^S_*/ϵ^2, which at time T_0(ϵ) = ϵ^−2T_*(ϵ)^1−b (b>0 small), satisfies smallness and decay conditions, compatible with those allowing one to prove global existence for the similar problem when the nonlinearity satisfies the null condition. Moreover, we have an asymptotic description of u close to the unique blowing up point (T_*(ϵ),0).
ER -