On the Cauchy problem for semilinear regularity-loss-type $\sigma$-evolution models with memory term

Abstract

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