On the Diffusion Phenomenon of Quasilinear Hyperbolic Waves

Abstract

The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping $$u_{tt}+u_t-{\rm div}(a(\nabla_u)\nabla_u)=0,$$ and show that, at least when n ≤ 3, they tend, as t → +∞, to those of the nonlinear parabolic equation $$v_t-{\rm div}(a(\nabla_v)\nabla_v)=0,$$ in the sense that the norm $\|u(.,t)-v(.,t)\|{_L\infty}({\rm R}^n)$ 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 Hsiao, L. and Liu Taiping (see [1, 2]).