## On the diffusion phenomenon of quasilinear hyperbolic waves.(English)Zbl 0959.35022

Summary: The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping $$u_{tt}+u_t-\text{div}(a(\nabla u)\nabla u)=0$$, and show that, at least when $$n\geq 3$$, they tend, as $$t\to +\infty$$, to those of the nonlinear parabolic equation $$v_t-\text{div}(a(\nabla v)\nabla v)=0$$, in the sense that the norm $$\|u(.,t)-v(.,t)\|_{L^\infty(\mathbb{R}^n)}$$ 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 T. Liu [Commun. Math. Phys. 143, No. 3, 599-605 (1992; Zbl 0763.35058); Chin. Ann. Math. Ser. B 14, No. 4, 46-40 (1993; Zbl 0804.35072)].

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35L70 Second-order nonlinear hyperbolic equations 35K55 Nonlinear parabolic equations 35L15 Initial value problems for second-order hyperbolic equations

### Citations:

Zbl 0763.35058; Zbl 0804.35072
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