Decay estimates of solutions for mildly degenerate Kirchhoff type dissipative wave equations in unbounded domains

Abstract

We consider the Cauchy problem for mildly degenerate Kirchhoff type dissipative wave equations: ρu″+‖A^1/2u(t)‖^2γAu+u′=0 in R^N×R⁺, u(x,0)=u₀(x), u′(x,0)=u₁(x) in R^N, with ρ>0, γ≥1, and ‖A^1/2u₀‖>0. When either the coefficient ρ or the initial data {u₀,u₁} are small, we prove the existence of global solutions by using several identities for energies. Moreover, we derive lower decay estimates of the solutions, and upper decay estimates of their second order derivatives.