Effect of additional regularity for the initial data on semi-linear $\sigma$-evolution equations with different damping types

TL;DR

This paper investigates how additional regularity in initial data influences the critical exponent and solution behavior of semi-linear σ-evolution equations with various damping types, revealing dependence of solution lifespan and blow-up phenomena.

Contribution

It introduces the impact of initial data regularity on the critical exponent and solution dynamics for semi-linear σ-evolution equations with different damping mechanisms.

Findings

Existence of global solutions for small initial data in subcritical cases.

Finite-time blow-up for supercritical initial data.

Lifespan estimates for solutions near blow-up conditions.

Abstract

In this paper, we would like to study the critical exponent for semi-linear $\sigma$-evolution equations with different damping types under the influence of additional regularity for the initial data. On the one hand, we establish the existence of global (in time) solutions for small initial data and the blow-up in finite time solutions in the supercritical case and the subcritical case, respectively. The very interesting phenomenon is that the critical case belonging to the global solution range or the blow-up solution range depends heavily on the assumption of additional regularity for the initial data. Furthermore, we are going to provide lifespan estimates for solutions when the blow-up phenomenon occurs.