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: $T(\epsilon )\le C\exp ({\epsilon }^{-2p(p-1)})$ when $p=p_S(n+\mu )$ for $0<\mu <\frac{n^2+n+2}{n+2}$. This result completes our previous study \cite{Tu-Lin} on the sub-Strauss type exponent $p<p_S(n+\mu )$. Our novelty is to construct the suitable test function from the modified Bessel function. This approach might be also applied to the other type damping wave equations.

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