On the diffusion phenomenonof quasilinear hyperbolic waves

We 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 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 L. Hsiao and Tai-ping Liu.