Decay characterization of solutions to semi-linear structurally damped σ-evolution equations with time-dependent damping

In this paper, we study the Cauchy problem to the linear damped σ-evolution equation with time-dependent damping in the effective cases {equation*} ut t+(-Δ)^σ u+b(t)(-Δ)^δ u_t=0, {equation*} and investigate the decay rates of the solution and its derivatives that are expressed in terms of the decay character of the initial data u₀(x)=u(0, x) and u₁(x)=uₜ(0, x). We are interested also in the existence and decay rate of the global in time solution with small data for the corresponding semi-linear problem with the nonlinear term of power type ||D|^γ u|ᵖ. The blow-up results for solutions to the semi-linear problem in the case γ=0 are presented to show the sharpness of the exponent p.