Resolving the one-dimensional autonomous flow-free explosion problem

We introduce the explosion problem for semilinear elliptic and parabolic PDEs. The explosion problem is a question of existence of nonnegative solutions to −Lφ = λg(x, φ) and nonnegative global solutions to ∂tφ−Lφ = λg(x, t, φ). Under certain conditions there is a λ∗ > 0, called the explosion threshold, for which λ < λ∗ implies a solution exists, and λ > λ∗ implies no solution exists. In this paper we focus on the one-dimensional elliptic case. Our main result is the explicit calculation of the explosion threshold λ∗ for the one-dimensional autonomous flow-free problem −φ′′ = λg(φ). Acknowledgment: We thank the National Science Foundation (grant DMS-1252912) and the Alfred P. Sloan foundation for their support.