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Existence of nonnegative solutions of nonlinear fractional parabolic inequalities

Existence of nonnegative solutions of nonlinear fractional parabolic inequalities We study the existence of nontrivial nonlocal nonnegative solutions u(x, t) of the nonlinear initial value problems (∂t-Δ)αu≥uλinRn×R,n≥1u=0inRn×(-∞,0)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} (\partial _t -\Delta )^\alpha u\ge u^\lambda &{}\quad \mathrm{in}\,\, {\mathbb {R}}^n \times {\mathbb {R}},\,n\ge 1 \\ u=0 &{}\quad \mathrm{in}\,\, {\mathbb {R}}^n \times (-\infty ,0) \end{array}\right. \end{aligned}$$\end{document}and C1uλ≤(∂t-Δ)αu≤C2uλinRn×R,n≥1u=0inRn×(-∞,0),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} C_1 u^\lambda \le (\partial _t -\Delta )^\alpha u\le C_2 u^\lambda &{}\quad \mathrm{in}\,{\mathbb {R}}^n \times {\mathbb {R}},\,n\ge 1 \\ u=0 &{}\quad \mathrm{in}\, {\mathbb {R}}^n \times (-\infty ,0), \end{array}\right. \end{aligned}$$\end{document}where λ,α,C1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda ,\alpha ,C_1$$\end{document}, and C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C_2$$\end{document} are positive constants with C1<C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C_1 <C_2$$\end{document}. We use the definition of the fractional heat operator (∂t-Δ)α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\partial _t -\Delta )^\alpha $$\end{document} given in Taliaferro (J Math Pures Appl 133:287–328, 2020) and compare our results in the classical case α=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha =1$$\end{document} to known results. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Existence of nonnegative solutions of nonlinear fractional parabolic inequalities

Journal of Evolution Equations , Volume 21 (4) – Dec 1, 2021

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-021-00739-6
Publisher site
See Article on Publisher Site

Abstract

We study the existence of nontrivial nonlocal nonnegative solutions u(x, t) of the nonlinear initial value problems (∂t-Δ)αu≥uλinRn×R,n≥1u=0inRn×(-∞,0)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} (\partial _t -\Delta )^\alpha u\ge u^\lambda &{}\quad \mathrm{in}\,\, {\mathbb {R}}^n \times {\mathbb {R}},\,n\ge 1 \\ u=0 &{}\quad \mathrm{in}\,\, {\mathbb {R}}^n \times (-\infty ,0) \end{array}\right. \end{aligned}$$\end{document}and C1uλ≤(∂t-Δ)αu≤C2uλinRn×R,n≥1u=0inRn×(-∞,0),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} C_1 u^\lambda \le (\partial _t -\Delta )^\alpha u\le C_2 u^\lambda &{}\quad \mathrm{in}\,{\mathbb {R}}^n \times {\mathbb {R}},\,n\ge 1 \\ u=0 &{}\quad \mathrm{in}\, {\mathbb {R}}^n \times (-\infty ,0), \end{array}\right. \end{aligned}$$\end{document}where λ,α,C1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda ,\alpha ,C_1$$\end{document}, and C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C_2$$\end{document} are positive constants with C1<C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C_1 <C_2$$\end{document}. We use the definition of the fractional heat operator (∂t-Δ)α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\partial _t -\Delta )^\alpha $$\end{document} given in Taliaferro (J Math Pures Appl 133:287–328, 2020) and compare our results in the classical case α=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha =1$$\end{document} to known results.

Journal

Journal of Evolution EquationsSpringer Journals

Published: Dec 1, 2021

Keywords: Fully fractional heat operator; Nonlocal solution; Initial value problem; Positive solution; Critical exponent; 35B09; 35B33; 35K58; 35R11

References