Semilinear $$\sigma $$-evolution models with friction and visco-elastic type damping

Abstract In this paper we study the global (in time) existence of small data Sobolev solutions to Cauchy problem for semilinear $$\sigma $$ σ -evolution models with friction and visco-elastic damping a power nonlinearity, namely, $$\begin{aligned} \left\{ \begin{array}{ll} u_{tt}+ (-\Delta )^\sigma u + u_t +(- \Delta u_t=\big ||D|^au\big |^p,\\ u(0,x)=0,\quad u_{t}(0,x)=u_1(x),\end{array} \right. \end{aligned}$$ u tt + ( - Δ ) t = | D a p , 0 x 1 where \ge 1$$ ≥ , $$p>1$$ > $$u_1\in L^m(\mathbb {R}^n) \cap H^{s-2\sigma }_q(\mathbb ∈ L m R n ∩ H q s 2 $$s\ge 2\sigma $$q\in (1,\infty )$$ ∞ $$m\in [1,q)$$ [ . nonlinearity suppose $$a\in [0,2\sigma We are interested in connections between regularity assumptions admissible range exponents p nonlinearity.