Asymptotic stability of diffusion wave for a semilinear wave equation with damping

This paper studies the asymptotic behavior of the solution to the Cauchy problem of a semilinear wave equation with damping vtt+vt+f(Dv)=Δv, x∈Rn, under some smallness conditions. By applying elementary energy method, we prove the solution of the above equation tends to the planar diffusion wave v¯(x11+t) time-asymptotically, where v¯(x11+t) is a self-similar solution of the one dimensional equation v¯t+C0v¯x12=v¯x1x1,v¯(±∞,t)=v±,v+≠v−, with C0=12∂2f(ξ)∂ξ12|ξ=0. In addition, this paper gives the L∞ time decay rate, namely, ‖v−v¯‖L∞=O(1)ε2(1+t)−γ4, where γ=min⁡{3,n}.