Existence of nonnegative solutions of nonlinear fractional parabolic inequalities

Steven D. Taliaferro

doi:10.1007/s00028-021-00739-6

Taliaferro, Steven D.

2021-12-01 00:00:00

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.

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Journal of Evolution Equations Springer Journals

http://www.deepdyve.com/lp/springer-journals/existence-of-nonnegative-solutions-of-nonlinear-fractional-parabolic-iLPh0m3Uvs

$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.$

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Existence of nonnegative solutions of nonlinear fractional parabolic inequalities

Existence of nonnegative solutions of nonlinear fractional parabolic inequalities

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Existence of nonnegative solutions of nonlinear fractional parabolic inequalities

Existence of nonnegative solutions of nonlinear fractional parabolic inequalities

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