A competition on blow-up for semilinear wave equations with scale-invariant damping and nonlinear memory term

TL;DR

This paper studies the conditions under which solutions to certain wave equations with damping and memory terms blow up, revealing a new competition effect influenced by the memory parameter.

Contribution

It introduces a new analysis of blow-up conditions for wave equations with scale-invariant damping and fractional memory, highlighting a novel competition effect for small memory parameters.

Findings

Identifies a new competition between damping and memory effects on blow-up.

Shows the influence of the fractional memory parameter on blow-up range.

Uses test function method and generalized Kato's lemma for analysis.

Abstract

In this paper, we investigate blow-up of solutions to semilinear wave equations with scale-invariant damping and nonlinear memory term in $\mathbb{R}^n$, which can be represented by the Riemann-Liouville fractional integral of order $1-\gamma$ with $\gamma\in(0,1)$. Our main interest is to study mixed influence from damping term and the memory kernel on blow-up conditions for the power of nonlinearity, by using test function method or generalized Kato's type lemma. We find a new competition, particularly for the small value of $\gamma$, on the blow-up range between the effective case and the non-effective case.