Blow-up for semilinear damped wave equations with sub-Strauss exponent in the scattering case

TL;DR

This paper proves finite-time blow-up for semilinear damped wave equations with sub-Strauss exponent in the scattering case, extending previous results to a broader damping range using novel multiplier techniques.

Contribution

It introduces a new multiplier method to establish blow-up results for damped wave equations with sub-Strauss exponent in the scattering regime.

Findings

Blow-up occurs for sub-Strauss exponent in the scattering case.

The new multiplier technique effectively analyzes damping effects.

Results extend previous blow-up results to a wider damping range.

Abstract

It is well-known that the critical exponent for semilinear damped wave equations is Fujita exponent when the damping is effective. Lai, Takamura and Wakasa in 2017 have obtained a blow-up result not only for super-Fujita exponent but also for the one closely related to Strauss exponent when the damping is scaling invariant and its constant is relatively small,which has been recently extended by Ikeda and Sobajima. Introducing a multiplier for the time-derivative of the spatial integral of unknown functions, we succeed in employing the technics on the analysis for semilinear wave equations and proving a blow-up result for semilinear damped wave equations with sub-Strauss exponent when the damping is in the scattering range.