Global Well-Posedness and Analyticity of Generalized Porous Medium Equation in Fourier-Besov-Morrey Spaces with Variable Exponent

Muhammad Zainul Abidin,Jiecheng Chen

doi:10.3390/math9050498

Abstract

In this paper, we consider the generalized porous medium equation. For small initial data u0 belonging to the Fourier-Besov-Morrey spaces with variable exponent, we obtain the global well-posedness results of generalized porous medium equation by using the Fourier localization principle and the Littlewood-Paley decomposition technique. Furthermore, we also show Gevrey class regularity of the solution.

Highlights

We consider the three dimensional generalized porous medium (GPM) equation: (ut + μΛ β u = ∇ · (u∇ Pu) in R3 × R+, u( x, 0) = u0Morrey Spaces with Variable in R3, (1)Exponent
Inspired by the above works, we obtain the global well-posedness and analyticity for Equation (1) and show the Gevery class regularity of the solution in homogenous FourierBesov-Morrey spaces with variable exponent by considering ∇K ∈ L1
For detailed study related to Besov spaces with variable exponent and Besov-Morrey spaces with variable exponent, we refer the reader to [32,33,34,35,36,37] and the references therein

Summary

Introduction

We consider the three dimensional generalized porous medium (GPM) equation:. Exponent. For more details in this direction we refer the reader to [1,9] and the references therein Another similar model appears, which explains a aggregation phenomena and collective motion in mechanics of continuous media and biology [10,11]. Established the local well-posedness for large initial data in Besov spaces and the global solution for small inital data. They given a blowup criterion for the solution. Inspired by the above works, we obtain the global well-posedness and analyticity for Equation (1) and show the Gevery class regularity of the solution in homogenous FourierBesov-Morrey spaces with variable exponent by considering ∇K ∈ L1. For detailed study related to Besov spaces with variable exponent and Besov-Morrey spaces with variable exponent, we refer the reader to [32,33,34,35,36,37] and the references therein

Preliminaries

The Well-Posedness

Gevrey Class Regularity

Conclusions