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

Abstract

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