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

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 $\mu\>1$.