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Classification of Solutions to Higher Fractional Order Systems

Authors
  • Le, Phuong1
  • 1 University of Economics and Law, Ho Chi Minh City, Vietnam Vietnam National University, Ho Chi Minh City, Vietnam , Ho Chi Minh City (Vietnam)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Jun 01, 2021
Volume
41
Issue
4
Pages
1302–1320
Identifiers
DOI: 10.1007/s10473-021-0417-5
Source
Springer Nature
Keywords
License
Yellow

Abstract

Let 0 < α, β < n and f, g ∈ C([0, ∞) × [0, ∞)) be two nonnegative functions. We study nonnegative classical solutions of the system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^{\tfrac{\alpha }{2}}}u = f(u,v)}&{\text{in}\;{\mathbb{R}^n},} \\ {{{( - \Delta )}^{\tfrac{\beta }{2}}}v = g(u,v)}&{\text{in}\;{\mathbb{R}^n},} \end{array}} \right.$$\end{document} and the corresponding equivalent integral system. We classify all such solutions when f(s, t) is nondecreasing in s and increasing in t, g(s, t) is increasing in s and nondecreasing in t, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{f({\mu ^{n - \alpha }}s,{\mu ^{n - \beta }}t)} \over {{\mu ^{n + \alpha }}}}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{g({\mu ^{n - \alpha }}s,{\mu ^{n - \beta }}t)} \over {{\mu ^{n + \beta }}}}$$\end{document} are nonincreasing in μ > 0 for all s, t ≥ 0. The main technique we use is the method of moving spheres in integral forms. Since our assumptions are more general than those in the previous literature, some new ideas are introduced to overcome this difficulty.

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