A Note on Semi-linear Wave Equations

TL;DR

This paper demonstrates the existence of solutions with arbitrarily large initial energy and specified lifespan for a broad class of semi-linear wave equations on three-dimensional space.

Contribution

It establishes that semi-linear wave equations on R^3 can have solutions with predetermined lifespan and initial energy, extending understanding of solution behavior in nonlinear wave dynamics.

Findings

Solutions exist with lifespan [0, T_0] for any T_0 > 0.

Initial energy can be made arbitrarily large, at least E_0.

The result applies to a large class of semi-linear wave equations.

Abstract

Inspired by the work of Wang and Yu [21] on wave maps, we show that for all positive numbers T_{0} > 0 and E_{0} > 0, a large kind of semi-linear wave equation on R \times R^{3} has a solution whose life-span is [0; T_{0}], and the energy of the initial Cauchy data is at least E_{0}.