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

Jia Wei He,Bashir Ahmad,Yong Zhou,Li Peng

doi:10.1515/anona-2021-0211

Abstract

Abstract We are devoted to the study of a semilinear time fractional Rayleigh-Stokes problem on ℝ N , 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.

Highlights

We are devoted to the study of a semilinear time fractional Rayleigh-Stokes problem on RN, which is derived from a non-Newtonain uid for a generalized second grade uid with Riemann-Liouville fractional derivative
Fractional calculus has proved a powerful tool to describe the viscoelasticity of uids and anomalous di usion phenomena, such as the constitutive relationship of the uid models [21], basic random walk models [25], and so on
Mainly due to the nonlocal characteristic of the fractional derivative, there are some excellent works on stochastic processes driven by fractional Brownian motion [13] and on physical phenomena like inverse problems for heat equation [19] and memory e ect [2]

Summary

Introduction

Fractional calculus has proved a powerful tool to describe the viscoelasticity of uids and anomalous di usion phenomena, such as the constitutive relationship of the uid models [21], basic random walk models [25], and so on. Mainly due to the nonlocal characteristic of the fractional derivative, there are some excellent works on stochastic processes driven by fractional Brownian motion [13] and on physical phenomena like inverse problems for heat equation [19] and memory e ect [2] It is worth mentioning some solid works about time-fractional derivatives [1, 4, 7, 12, 14, 17, 37,38,39] and space-fractional derivatives [15, 16, 33] and the references therein, most of conclusions in these works commuted with fractional models are quite di erent from the situation of integer derivative, for instance, decay and asymptotical behaviors, blow-up analysis, well-posedness analysis, stability and Liouville property etc.

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