Life-span of semilinear wave equations with scale-invariant damping: Critical Strauss exponent case

Abstract

The blow up problem of the semilinear scale-invariant damping wave equation with critical Strauss type exponent is investigated. The life span is shown to be: $T(\varepsilon)\leqslant \exp(C\varepsilon^{-p(p-1)})$ when $p=p_S(n+\mu)$ for $0 < \mu < \frac{n^2+n+2}{n+2}$. This result completes our previous study [9] on the sub-Strauss type exponent $p < p_S(n+\mu)$. Different from the work of M. Ikeda and M. Sobajima [5], we construct the suitable test function by introducing the modified Bessel function of second type. We note this method can be easily extended to some other scale-invariant wave models even with the Laplacian of variable coefficients.