Life-Span of Semilinear Wave Equations with Scale-invariant Damping: Critical Strauss Exponent Case

TL;DR

This paper investigates the lifespan of solutions to semilinear wave equations with scale-invariant damping at the critical Strauss exponent, providing an exponential upper bound and introducing a novel test function approach.

Contribution

It establishes the lifespan estimate at the critical Strauss exponent using a new test function derived from the modified Bessel function, extending previous sub-Strauss results.

Findings

Lifespan T(ε) ≤ C exp(ε^{-2p(p-1)}) at critical exponent p=p_S(n+μ)

Constructed a new test function from modified Bessel functions

Extended previous results on sub-Strauss exponents

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: $T(\varepsilon)\leq C\exp(\varepsilon^{-2p(p-1)})$ 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.