Global existence for semilinear wave equations with the critical blow-up term in high dimensions

TL;DR

This paper investigates conditions for global existence of solutions to semilinear wave equations with critical blow-up terms in high dimensions, providing new examples and insights into the criteria distinguishing between almost global and global solutions.

Contribution

The paper introduces a new example of global existence in four-dimensional wave equations with critical nonlinear terms, advancing understanding of the criteria for solution longevity.

Findings

Provided an example of global existence in four dimensions

Analyzed nonlinear integral terms related to derivative loss

Contributed to the criterion development for solution existence

Abstract

We are interested in almost global existence cases in the general theory for nonlinear wave equations, which are caused by critical exponents of nonlinear terms. Such situations can be found in only three cases in the theory, cubic terms in two space dimensions, quadratic terms in three space dimesions and quadratic terms including a square of unknown functions itself in four space dimensions. Except for the last case, criterions to classify nonlinear terms into the almost global, or global existence case, are well-studied and known to be so-called null condition and non-positive condition. Our motivation of this work is to find such a kind of the criterion in four space dimensions. In our previous paper, an example of the non-single term for the almost global existence case is introduced. In this paper, we show an example of the global existence case. These two examples have nonlinear integral terms which are closely related to derivative loss due to high dimensions. But it may help us to describe the final form of the criterion.