A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent

TL;DR

This paper extends the blowup analysis of scale invariant damping wave equations with sub-Strauss exponent, demonstrating wave-like behavior and providing lifespan estimates without restrictions on the damping parameter.

Contribution

It broadens the blowup exponent range and derives uniform lifespan bounds, extending previous results to larger damping parameters without smallness restrictions.

Findings

Extended blowup exponent range to 1<p<p_S(n+μ)

Derived uniform lifespan estimate T(ε) ≤ Cε^{-2p(p-1)/γ(p,n+2μ)}

Showed wave-like behavior persists with large μ>1

Abstract

We concern the blow up problem to the scale invariant damping wave equations with sub-Strauss exponent. This problem has been studied by Lai, Takamura and Wakasa (\cite{Lai17}) and Ikeda and Sobajima \cite{Ikedapre} recently. In present paper, we extend the blowup exponent from $p_F(n)\leq p<p_S(n+2\mu)$ to $1<p<p_S(n+\mu)$ without small restriction on $\mu$. Moreover, the upper bound of lifespan is derived with uniform estimate $T(\varepsilon)\leq C\varepsilon^{-2p(p-1)/\gamma(p,n+2\mu)}$. This result extends the blowup result of semilinear wave equation and shows the wave-like behavior of scale invariant damping wave equation's solution even with large $\mu>1$.