Nonexistence of solutions to fractional parabolic problem with general nonlinearities

Abstract

In this content, we investigate a class of fractional parabolic equation with general nonlinearities ∂z(x,t)∂t-(Δ+λ)β2z(x,t)=a(x1)f(z),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\partial z(x,t)}{\\partial t}-(\\Delta +\\lambda )^{\\frac{\\beta }{2}}z(x,t)=a(x_{1})f(z), \\end{aligned}$$\\end{document}where a and f are nondecreasing functions. We first prove that the monotone increasing property of the positive solutions in x1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_{1}$$\\end{document} direction. Based on this, nonexistence of the solutions are obtained by using a contradiction argument. We believe these new ideas we introduced will be applied to solve more fractional parabolic problems.