Stability of planar diffusion wave for the quasilinear wave equation with nonlinear damping
Tóm tắt
In this paper, we will show that under some smallness conditions, the planar diffusion wave
$$\bar v\left( {\frac{{x_1 }}
{{\sqrt {1 + t} }}} \right)$$
is stable for a quasilinear wave equation with nonlinear damping: v
tt
− Δf(v) + v
t
+ g(v
t
) = 0, x = (x
1, x
2, ⋯, x
n
) ∈ ℝ
n
, where
$$\bar v\left( {\frac{{x_1 }}
{{\sqrt {1 + t} }}} \right)$$
is the unique similar solution to the one dimensional nonlinear heat equation:
$v_t - f(v)_{x_1 x_1 } = 0,f'(v) > 0$
, v(±∞, t) = v
±, v
+ ≠ v
−. We also obtain the L
∞ time decay rate which reads
$\left\| {v - \bar v} \right\|_{L^\infty } = O(1)(1 + t) - \tfrac{r}
{4}
$
, where r = min{3, n}. To get the main result, the energy method and a new inequality have been used.
Tài liệu tham khảo
Dafermos, C. A system of hyperbolic conservation laws with frictional damping. Z. Angew. Math. Phys., 46(Special Issue): 294–307 (1995)
He, C., Huang, F., Yong, Y. Stability of planar diffusion wave for nonlinear evolution equation. Preprint
Huang, F., Li, J., Matsmura, A. Stability of combination of viscous contact waves with rarefaction waves for 1-D compressible Navier-Stokes system. Arch. Rat. Mech. Anal., 2010, online DOI: 10.1007/s00205-009-0267-0
Hsiao, L., Liu, T. Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Commun. Math. Phys., 143: 599–605 (1992)
Hsiao, L., Liu, T. Nonlinear diffusion phenomena of nonlinear hyperbolic system. Chinese Ann. Math. Ser. B, 14: 465–480 (1993).
Hsiao, L., Pan, R. Initial boundary value problem for the system compressible adiabatic flow through porous media. J. Diff. Eqns., 159: 280–305 (1999)
Hsiao, L., Luo, T. Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media. J. Diff. Eqns., 125: 329–365 (1996)
Liu, T., Wang, W. The pointwise estimates for diffusion wave for the Navier-Stokes systems in odd multidimensions. Comm. Math. Phys., 196: 145–173 (1998)
Liu, Y., Wang, W. The pointwise estimates of solutions for dissipative wave equation in multi-dimensions. Discrete Contin. Dyn. Syst., 20: 1013–1028 (2008)
Marcati, P., Milani, A. The one-dimensional Darcy’s law as the limit of a compressible Euler flow. J. Diff. Eqns., 84: 129–147 (1990)
Marcati, P., Mei, M., Rubino, B. Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping. J. Math. Fluid Mech., 7: 224–240 (2005)
Matsumura, A. On the asymptotic behavior of solutions of semi-linear wave equation. Publ. RIMS, Kyoto Univ., 12: 169–189 (1976)
Matsumura, A. Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equation with the first-order dissipation. Publ. RIMS, Kyoto Univ., 13: 349–379 (1977)
Nishihara, K. Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping. J. Diff. Eqns., 131: 171–188 (1996)
Nishihara, K. Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping. J. Diff. Eqns., 137: 384–395 (1997)
Nishihara, K., Yang, T. Boundary effect on asymptotic behavior of solutions to the p-system with linear damping. J. Diff. Eqns., 156: 439–458 (1999)
Nishihara, K., Wang, W., Yang, T. L p convergence rate to nonlinear diffusion waves for p-system with damping. J. Diff. Eqns., 161: 191–218 (2000)
Pan, R. Darcy’s law as long-time limit of adiabatic porous media flow. J. Diff. Eqns., 220: 121–146 (2006)
Zhao, H. Convergence rate to strong nonlinear diffusion waves for solutions of p-system with damping. J. Diff. Eqns., 174: 200–236 (2001)
Zhu, C., Jiang, M. L p decay rates to nonlinear diffusion waves for p-system with nonlinear damping. Science in China, Series A, 49: 721–739 (2006)
Zhu, C., Jiang, M. Convergence rates to nonlinear diffusion waves for p-system with nonlinear damping on quadrant. Discrete and Continuous Dynamical Systems, 23(3): 887–918 (2009)