## 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

### Keywords:

diffusion phenomenon; linear damping
Full Text:

### References:

 [1] Hsiao, L.; Liu, Tai-ping, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservations with damping, Comm. math. phys., Vol. 143, 599-605, (1992) · Zbl 0763.35058 [2] Hsiao, L.; Liu, Tai-ping, Nonlinear diffusive phenomena of nonlinear hyperbolic systems, Chin. ann. math., Vol. 14(B), 465-480, (1993) · Zbl 0804.35072 [3] Klainerman, S.; Ponce, G., Global, small amplitude solutions to nonlinear evolution equations, Comm. pure appl. math., Vol. 37, 443-455, (1984) [4] Li, Ta-tsien, Nonlinear heat conduction with finite speed of propagation, () · Zbl 0887.35097 [5] Li, Ta-tsien; Chun, Yun-mei, () [6] Nishihara, K., Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with linear damping, J. differential equations, Vol. 131, 171-188, (1996) · Zbl 0866.35066 [7] Nishihara, K., Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping, J. differential equations, Vol. 137, 384-395, (1997) · Zbl 0881.35076 [8] Zheng, Song-mu; Chen, Yun-mei, Global existence for nonlinear parabolic equations, Chin. ann. math., Vol. 7B, 57-73, (1986) · Zbl 0603.35049 [9] Li, Ya-chun, Classical solutions to fully nonlinear wave equations with dissipation, Chin. ann. math., Vol. 17A, 451-466, (1996) · Zbl 0926.35092 [10] Matsumura, A., On the asymptotic behavior of solutions of semilinear wave equations, Publ. RIMS Kyoto univ., Vol. 121, 169-189, (1976) · Zbl 0356.35008 [11] Racke, R., Lectures on nonlinear evolution equations, (1992), Vieweg · Zbl 0798.35107
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.