On the diffusion phenomenonof quasilinear hyperbolic waves

Abstract

We consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping utt+ut−div(a(∇u)∇u)=0, and show that they tend, as t→+∞, to those of the nonlinear parabolic equation vt−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.