A blow-up result for semi-linear structurally damped $\sigma$-evolution equations

TL;DR

This paper establishes a blow-up result for semi-linear structurally damped $\sigma$-evolution equations with fractional Laplacians, introducing new test functions to handle non-local operators.

Contribution

It provides a novel approach with new test functions to prove blow-up for fractional order damped evolution equations, extending existing methods.

Findings

Proved blow-up for fractional $\sigma$-evolution equations.

Developed new test functions for non-local operators.

Extended blow-up results to fractional damping cases.

Abstract

We would like to prove a blow-up result for semi-linear structurally damped $\sigma$-evolution equations, where $\sigma \ge 1$ and $\delta\in [0,\sigma)$ are assumed to be any fractional numbers. To deal with the fractional Laplacian operators $(-\Delta)^\sigma$ and $(-\Delta)^\delta$ as well-known non-local operators, in general, it seems difficult to apply the standard test function method directly. For this reason, in this paper we shall construct new test functions to overcome this difficulty.