Existence of solutions of semilinear wave equations with time-dependent propagation speed and time derivative nonlinearity
Kimitoshi Tsutaya, Yuta Wakasugi
TL;DR
This work analyzes semilinear wave equations with time-dependent propagation speed $a(t)$ and a time-derivative nonlinearity, formulated as $u_{tt}-a(t)^2\Delta u+b(t)u_t=F(u,u_t,\nabla_x u)$ with $F$ including $|u_t|^p$. Using energy methods and a fixed-point framework in the energy space, the authors establish global existence for the critical case $F(u_t)=|u_t|^p$ under weaker assumptions on $a$ and $b$, notably showing global small-data solutions when $a^{p-1}\in L^1([0,\infty))$ (yielding global existence in $n=1,2$ for all $p>1$), and derive Lifespan bounds when this integrability fails. They also prove finite-time blow-up for the time-derivative nonlinearity in contracting-universe scenarios, providing precise upper bounds on the lifespan $T_\varepsilon$ that depend on the growth of $a(t)$ and $b_-(t)$, including AdS-type coefficients. The results connect to FLRW, de Sitter, and anti-de Sitter spacetimes, offering critical exponents and lifespan estimates that distinguish expanding vs contracting cosmologies and highlight the role of integrability of $a^{p-1}$ in global behavior.
Abstract
Consider wave equations with time derivative nonlinearity and time-dependent propagation speed which are generalized versions of the wave equations in the Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime, the de Sitter spacetime and the anti-de Sitter space time. We show lower bounds of the lifespan of solutions as well as the global existence by providing an integrability condition on the propagation speed function, which is applicable to the nonlinear wave equation in the expanding FLRW spacetime including the de Sitter spacetime. We also prove that blow-up in a finite time occurs for the generalized form of the equation in contracting universes such as the anti-de Sitter spacetime, as well as upper bounds of the lifespan of blow-up solutions.