Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations

Abstract

We study the generalized Rayleigh-Stokes problem involving a fractional derivative and nonlinear perturbation. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and asymptotic stability of solutions. In particular, if the nonlinearity is Lipschitzian then the mild solution of the mentioned problem becomes a classical one and its convergence to equilibrium point is proved.

On global solvability and regularity for generalized Rayleigh-Stokes equations with history-dependent nonlinearities

“…This enables us to consider the case when f contains polymonial or gradient terms, which have connections with practical applications. This also extends the recent results established in [12][13][14].…”

“…It should be mentioned that, some numerical schemes for Rayleigh-Stokes equations were developed in [1,2,5,6]. On the other hand, analytical representations for solution of (1.4) in linear case were obtained in [11,19], and recently, the regularity for nonlinear Rayleigh-Stokes equations has been established in [13,14,21]. For more studies related to (1.4), we refer the reader to [15,17,20], where the terminal value problem was carried out.…”

On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations

“…It should be mentioned that, the obtained results can be applied to the problems governed by nonclassical diffusion equation, the semilinear Rayleigh-Stokes equation and the Basset equation. In particular, our results extend those given in [12]. Moreover, in comparison with the studies in [11,22] for subdiffusion equations, the decay rate in our stability results established for (1.1) is more explicit and faster (see Remark 2).…”

On nonlinear Sobolev equation with the Caputo fractional operator and exponential nonlinearity

“…The control of exponential nonlinearity is our biggest challenge in this paper. The main reason is that, unlike in [34, 35] articles, we cannot evaluate directly in $$$ {L}^p $$$ space. Therefore, we have to study some more techniques as well as some important inequalities in several papers like [32, 33].…”