Fujita versus Strauss - a never ending story

TL;DR

This paper establishes blow-up and non-existence results for solutions to a semi-linear wave equation with scale-invariant damping and mass, using variable transformations, Kato's lemma, and explicit integral formulas, especially focusing on the critical exponent case.

Contribution

It introduces a novel approach combining variable changes and Kato's lemma to analyze blow-up in a wave equation with scale-invariant terms, including the critical exponent case.

Findings

Blow-up occurs for solutions with certain initial data support and sign conditions.

Explicit integral representation for the linear problem in 1D is derived at the critical exponent.

Non-existence of global solutions is proved in the critical case.

Abstract

In this paper, we obtain a blow-up result for solutions to a semi-linear wave equation with scale-invariant dissipation and mass and power non-linearity, in the case in which the model has a "wave like" behavior. In order to achieve this goal, we perform a change of variables that transforms our starting equation in a strictly hyperbolic semi-linear wave equation with time-dependent speed of propagation. Then, we apply Kato's lemma to find a blow-up result for solutions to the transformed equation under some support and sign assumptions on the initial data. A special emphasis is placed on the limit case, that is, when the exponent p is exactly equal to the upper bound of the range of admissible values of p for which this blow-up result is valid. In this critical case an explicit integral representation formula for solutions of the corresponding linear Cauchy problem in 1d is derived. Finally, carrying out the inverse change of variables we get a non-existence result for global (in time) solutions to the original model.