We will look at reaction–diffusion type equations of the following type, ∂tβV(t,x)=-(-Δ)α/2V(t,x)+It1-β[V(t,x)1+η].\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial ^\\beta _tV(t,x)=-(-\\Delta )^{\\alpha /2} V(t,x)+I^{1-\\beta }_t[V(t,x)^{1+\\eta }]. \\end{aligned}$$\\end{document}We first study the equation on the whole space by making sense of it via an integral equation. Roughly speaking, we will show that when 0<eta leqslant eta _c, there is no global solution other than the trivial one while for eta >eta _c, non-trivial global solutions do exist. The critical parameter eta _c is shown to be frac{1}{eta ^*} where η∗:=supa>0supt∈(0,∞),x∈Rdta∫RdG(t,x-y)V0(y)dy<∞\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\eta ^*:=\\sup _{a>0}\\left\\{ \\sup _{t\\in (0,\\,\\infty ),x\\in \\mathbb {R}^d}t^a\\int _{\\mathbb {R}^d}G(t,\\,x-y)V_0(y)\\,\\mathrm{d}y<\\infty \\right\\} \\end{aligned}$$\\end{document}and G(t,,x) is the heat kernel of the corresponding unforced operator. V_0 is a non-negative initial function. We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the fractional time derivative induces a significant change in the behavior of the solution.