The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension, I

Abstract

In this paper, we consider mixed problems with a spacelike boundary derivative condition for semilinear wave equations with exponential nonlinearities in a quarter plane. Results similar to those obtained earlier by Caffarelli-Friedman for Cauchy problems and power nonlinearities are proved in the present situation, namely we show that solutions either are global or blow up on a $C^1$ spacelike curve. Weaker results are also obtained if the boundary vector field is tangent to the characteristic which leaves the domain in the future.