Fujita exponent for non-local parabolic equation involving the Hardy–Leray potential

Abstract

In this paper, we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy–Leray-type potential. More precisely, we consider the problem (wt-Δw)s=λ|x|2sw+wp+f,inRN×(0,+∞),w(x,t)=0,inRN×(-∞,0],\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} (w_t-\\Delta w)^s=\\frac{\\lambda }{|x|^{2s}} w+w^p +f, &{}\\quad \ ext {in }\\mathbb {R}^N\ imes (0,+\\infty ),\\\\ w(x,t)=0, &{}\\quad \ ext {in }\\mathbb {R}^N\ imes (-\\infty ,0], \\end{array}\\right. } \\end{aligned}$$\\end{document}where N>2s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N> 2s$$\\end{document}, 0<s<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<s<1$$\\end{document} and 0<λ<ΛN,s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\lambda <\\Lambda _{N,s}$$\\end{document}, the optimal constant in the fractional Hardy–Leray inequality. In particular, we show the existence of a critical existence exponent p+(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_{+}(\\lambda , s)$$\\end{document} and of a Fujita-type exponent F(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(\\lambda ,s)$$\\end{document} such that the following holds:Let p>p+(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>p_+(\\lambda ,s)$$\\end{document}. Then there are not any non-negative supersolutions.Let p<p+(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p<p_+(\\lambda ,s)$$\\end{document}. Then there exist local solutions, while concerning global solutions we need to distinguish two cases:Let 1<p≤F(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 1< p\\le F(\\lambda ,s)$$\\end{document}. Here we show that a weighted norm of any positive solution blows up in finite time.Let F(λ,s)<p<p+(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(\\lambda ,s)<p<p_+(\\lambda ,s)$$\\end{document}. Here we prove the existence of global solutions under suitable hypotheses.