On well-posedness of semilinear Rayleigh-Stokes problem with fractional derivative on ℝN

Abstract

We are devoted to the study of a semilinear time fractional Rayleigh-Stokes problem on ℝN, which is derived from a non-Newtonain fluid for a generalized second grade fluid with Riemann-Liouville fractional derivative. We show that a solution operator involving the Laplacian operator is very effective to discuss the proposed problem. In this paper, we are concerned with the global/local well-posedness of the problem, the approaches rely on the Gagliardo-Nirenberg inequalities, operator theory, standard fixed point technique and harmonic analysis methods. We also present several results on the continuation, a blow-up alternative with a blow-up rate and the integrability in Lebesgue spaces.

Keywords

• fractional derivative
• rayleigh-stokes problem
• well-posedness
• integrability
• 26a33
• 34a12
• 35r11