Abstract
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 dampingsutt+(−Δ)σu+(−Δ)δ1ut+(−Δ)δ2ut=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 |ut|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⁎.
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