On the diffusion phenomenon of quasilinear hyperbolic waves

The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping U_tt + u_t -div (a(∇_u) ∇_u) = 0, and show that, at least when n ≤ 3, they tend, as t → +∞, to those of the nonlinear parabolic equation v_t - div (a( ∇_v) ∇_v) = 0, in the sense that the norm ∥u(.,t) - v(., t)∥L∞(Rⁿ) of the difference u - v decays faster than that of either u or v. This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves, first observed by Hsiao, L. and Liu Taiping (see [1, 2]).