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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
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