Fujita phenomena in nonlinear fractional Rayleigh-Stokes equations

TL;DR

This paper investigates the Fujita phenomena in nonlinear fractional Rayleigh-Stokes equations, establishing a critical exponent for solution blow-up that is independent of the fractional order, using decay estimates and test functions.

Contribution

It identifies the critical Fujita exponent for these equations and demonstrates its independence from the fractional derivative parameter, providing new analytical techniques.

Findings

Existence of a critical Fujita exponent for blow-up

Critical exponent is independent of fractional order α

Decay estimates are key to the analysis

Abstract

This paper concerns the Cauchy problems for the nonlinear Rayleigh-Stokes equation and the corresponding system with time-fractional derivative of order $\alpha\in(0,1)$, which can be used to simulate the anomalous diffusion in viscoelastic fluids. It is shown that there exists the critical Fujita exponent which separates systematic blow-up of the solutions from possible global existence, and the critical exponent is independent of the parameter $\alpha$. Different from the general scaling argument for parabolic problems, the main ingredients of our proof are suitable decay estimates of the solution operator and the construction of the test function.