Global existence and lifespan for semilinear wave equations with mixed nonlinear terms

TL;DR

This paper investigates the lifespan and global existence of solutions to semilinear wave equations with mixed nonlinearities, deriving optimal lifespan estimates and identifying conditions for global solutions and blow-up in various dimensions.

Contribution

It provides new lifespan estimates for equations with critical and supercritical nonlinearities and establishes conditions for global existence versus blow-up in coupled wave systems.

Findings

Optimal lifespan in 3D: ^{- ext{power}} growth.

Improved lower bounds for lifespan in 2D.

Global existence for certain nonlinear wave systems.

Abstract

Firstly, we study the equation $\square u = |u|^{q_c}+ |\partial u|^p$ with small data, where $q_c$ is the critical power of Strauss conjecture and $p\geq q_c.$ We obtain the optimal lifespan $\ln({T_\varepsilon})\approx\varepsilon^{-q_c(q_c-1)}$ in $n=3$, and improve the lower-bound of $T_\varepsilon$ from $\exp({c\varepsilon^{-(q_c-1)}})$ to $\exp({c\varepsilon^{-(q_c-1)^2/2}})$ in $n=2$. Then, we study the Cauchy problem with small initial data for a system of semilinear wave equations $\square u = |v|^q,$ $ \square v = |\partial_t u|^p$ in 3-dimensional space with $q<2$. We obtain that this system admits a global solution above a $p-q$ curve for spherically symmetric data. On the contrary, we get a new region where the solution will blow up.