In this paper we study the global (in time) existence of small data Sobolev solutions to the Cauchy problem for semilinear sigma -evolution models with friction and visco-elastic damping and with a power nonlinearity, namely,utt+(-Δ)σu+ut+(-Δ)σut=||D|au|p,u(0,x)=0,ut(0,x)=u1(x),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} u_{tt}+ (-\\Delta )^\\sigma u + u_t +(- \\Delta )^\\sigma u_t=\\big ||D|^au\\big |^p,\\\\ u(0,x)=0,\\quad u_{t}(0,x)=u_1(x),\\end{array} \\right. \\end{aligned}$$\\end{document}where sigma ge 1, p>1, and the data u1∈Lm(Rn)∩Hqs-2σ(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_1\\in L^m(\\mathbb {R}^n) \\cap H^{s-2\\sigma }_q(\\mathbb {R}^n) $$\\end{document}with sge 2sigma , qin (1,infty ) and min [1,q). In the power nonlinearity we suppose ain [0,2sigma ). We are interested in connections between regularity assumptions for the data and the admissible range of exponents p in the power nonlinearity.