Semilinear mixed problems in exterior domains for σ-evolution equations with friction and coefficients depending on spatial variables

The main purpose of this paper is to investigate decay estimates for solutions to the Cauchy problemutt+a1(x)(−Δ)σu+aut=0,u(0,x)=u0(x),ut(0,x)=u1(x)forx∈Rn, as well as the estimates for solutions to the corresponding Cauchy-Dirichlet problem in an exterior domain Ω⊂Rn. Here a is a positive constant. The coefficient a1=a1(x) is supposed to be continuous and positive on the closure Ω‾. The parameter σ∈(0,1) brings to the model the so-called Levi-stable behavior for the corresponding diffusion stochastic process. Finally, we show the global (in time) existence of energy solutions from evolution spaces to the semilinear modelsutt+a1(x)(−Δ)σu+aut=|ut|p,u(0,x)=u0(x),ut(0,x)=u1(x), in domain (0,∞)×Ω with arbitrarily small initial data.