Decay character and the semilinear structurally damped σ-evolution equations with two dissipative terms

In this paper, we study the decay rates of the solution and its derivatives to the Cauchy problem for the linear damped σ -evolution equation with two dissipative dampings u t t + ( − Δ ) σ u + ( − Δ ) δ 1 u t + ( − Δ ) δ 2 u t = 0 that are expressed in terms of decay characters of the initial data. Furthermore, by using the decay rates of solutions to the linear part and the fixed-point method, we investigate the existence and decay rate of the global solution with small data of the corresponding semi-linear problems with the nonlinear term of power type | u t | p , for the supercritical case p > p ⁎ . For some special data spaces, the optimality of p ⁎ is guaranteed by nonexistence results in the subcritical case p < p ⁎ and the critical case p = p ⁎ .