On the Cauchy problem for semilinear regularity-loss-type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si4.svg"><mml:mi>?</mml:mi></mml:math>-evolution models with memory term

In this paper, we consider the Cauchy problem for semilinear $\sigma$-evolution models with an exponential decay memory term. Concerning corresponding linear problem, derive some regularity-loss-type estimates of solutions and generalized diffusion phenomena. Particularly, obtained estimations are sharper than those in previous paper [20]. Then, determine critical exponents power nonlinearity spatial dimensions by proving global (in time) existence Sobolev low regularity fractional orders blow-up result even any value $\sigma\geqslant 1$.