Asymptotically self-similar global solutions of a damped wave equation with nonlinear memory

In this paper we consider the Cauchy problem of semi-linear damped wave equation with nonlinear memory term a∫₀^t(t−τ)^−γ|u|^α−1u(τ,x) dτ. We prove global existence and asymptotic behavior of solution for small initial data. Moreover, we show that the global solutions behave asymptotically like self-similar solutions of the semi-linear heat equation with nonlinear memory as t→∞.