Self‐similar asymptotic profile of the solution to a nonlinear evolution equation with critical dissipation

In this paper, we obtain optimal decay estimates for the solutions to an evolution equation with critical, structural, dissipation, and absorbing power nonlinearity:u t t +Δ2 θ u + 2 μ ( − Δ ) θ u t + | u t | p − 1u t = 0 , t ≥ 0 , x ∈ R n ,with  μ >0, θ is a positive integer, and  p >1+4 θ / n , in space dimension  n ∈(2 θ ,4 θ ). We use these estimates to find the self‐similar asymptotic profile of the solutions, when  μ ≥1.