Life-Span of Solutions to Critical Semilinear Wave Equations

TL;DR

This paper simplifies the proof of the sharp lifespan estimate for solutions to critical semilinear wave equations, confirming the optimality of lifespan bounds in high-dimensional cases with small initial data.

Contribution

It provides a simpler proof of Takamura and Wakasa's lifespan estimate using Zhou's method, resolving an open problem in the theory of nonlinear wave equations.

Findings

Established sharp upper bounds for solution lifespan in critical cases

Simplified proof technique applicable to high dimensions

Confirmed optimality of lifespan estimates in specific nonlinear wave scenarios

Abstract

The final open part of the famous Strauss conjecture on semilinear wave equations of the form \Box u=|u|^{p}, i.e., blow-up theorem for the critical case in high dimensions was solved by Yordanov and Zhang, or Zhou independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. Recently, Takamura and Wakasa have obtained the sharp upper bound of the lifespan of the solution to the critical semilinear wave equations, and their method is based on the method in Yordanov and Zhang. In this paper, we give a much simple proof of the result of Takamura and Wakasa by using the method in Y. Zhou for space dimensions n\geq 2. Simultaneously, this estimate of the life span also proves the last open optimality problem of the general theory for fully nonlinear wave equations with small initial data in the case n=4 and quadratic nonlinearity(One can see Li and Chen for references on the whole history).