On the diffusion phenomenonof quasilinear hyperbolic waves

Abstract 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 ∞ ( R n ) 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.