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On the diffusion phenomenon of quasilinear hyperbolic waves. (English) Zbl 0959.35126

Summary: We 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 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 Tai-ping Liu.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
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