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Stable weak solutions of weighted nonlinear elliptic equations
The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space
1. | Mathematics department, Shanghai Jiao Tong University, Shanghai 200240, China |
2. | Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, 200240, Shanghai |
References:
[1] |
L. L. Fan, H. X. Liu and H. Yin, Dacay estimates of planar stationary waves for damped wave equations with nonlinear convection in mutil-dimensional half space, Acta Math Sci, 31(B) (2011), 1389-1410.
doi: 10.1016/S0252-9602(11)60326-3. |
[2] |
T. Hosono and T. Ogawa, Large time behavior and $L^p-L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differ. Equations, 203 (2004), 82-118.
doi: 10.1016/j.jde.2004.03.034. |
[3] |
L. C. Evans, "Partial Differential Equations," Graduate in Math., 19, Amer. Math. Soc., Providence, RI, 1998. |
[4] |
T. Li and Y. M. Chen, "Global Classical Solutions for Nonlinear Evolution Equations," Pitman Monogr. Surv. Pure Appl. Math., vol. 45, Longman Scientific and Technical, Harlow, 1992. |
[5] |
T. P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math, 50 (1997), 1113-1182.
doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.3.CO;2-8. |
[6] |
T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Comm. Math. Phys., 169 (1998), 145-173.
doi: 10.1007/s002200050418. |
[7] |
T. P. Liu and Y. Zeng, Large time behavior of solutions general quasilinear hyperbolic-parabolic systems of conservation laws, A. M. S. memoirs, 599 (1997). |
[8] |
Y. Q. Liu, The point-wise estimates of solutions for semi-linear dissipative wave equation, Comm. Pure Appl. Anal., 12 (2013), 237-252.
doi: 10.3934/cpaa.2013.12.237. |
[9] |
Y. Q. Liu and W. K. Wang, The pointwise estimates of solutions for dissipative wave equation in multi-dimensions, Discrete Contin. Dyn. Syst., 20 (2008), 1013-1028.
doi: 10.3934/dcds.2008.20.1013. |
[10] |
M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semiliear dissipative wave equation, Math. Z., 214 (1993), 325-241.
doi: 10.1007/BF02572407. |
[11] |
K. Nishihara, Global asymptotics for the damped wace equation with absotption in higher dimensional space, J. Math. Soc., Japan, 58 (2006), 805-836. |
[12] |
K. Nishihara and H. J. Zhao, Dacay properties of solutions to the Cauchy problem for the damped wace equation with absorption, J. Math. Anal. Appl., 313 (2006), 698-610.
doi: 10.1016/j.jmaa.2005.08.059. |
[13] |
K. Ono, Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations, Discrete Contin. Dyn. Syst., 9 (2003), 651-662.
doi: 10.3934/dcds.2003.9.651. |
[14] |
R. Ikehata, A remark on a critical exponent for the semilinear dissipative wave equation in the one dimensional half space, Differential Integral Equations, 16 (2003), 727-736. |
[15] |
R. Ikehata, K. Nishihara and H. J. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differ. Equation, 226 (2006), 1-29.
doi: 10.1016/j.jde.2006.01.002. |
[16] |
S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (1995), 617-653.
doi: 10.2969/jmsj/04740617. |
[17] |
Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term, Adv. Math. Sci. Appl., 18, (2008), 329-343. |
[18] |
Y. Ueda, T. Nakamura and S. Kawashima, Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space, Kinet. Relat. Models, 1 (2008), 49-64.
doi: 10.3934/krm.2008.1.49. |
[19] |
W. K. Wang and W. J. Wang, The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions, J. Math. Anal. Appl., 366 (2010), 226-241.
doi: 10.1016/j.jmaa.2009.12.013. |
[20] |
W. K. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differ. Equations, 173 (2001), 410-450.
doi: 10.1006/jdeq.2000.3937. |
show all references
References:
[1] |
L. L. Fan, H. X. Liu and H. Yin, Dacay estimates of planar stationary waves for damped wave equations with nonlinear convection in mutil-dimensional half space, Acta Math Sci, 31(B) (2011), 1389-1410.
doi: 10.1016/S0252-9602(11)60326-3. |
[2] |
T. Hosono and T. Ogawa, Large time behavior and $L^p-L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differ. Equations, 203 (2004), 82-118.
doi: 10.1016/j.jde.2004.03.034. |
[3] |
L. C. Evans, "Partial Differential Equations," Graduate in Math., 19, Amer. Math. Soc., Providence, RI, 1998. |
[4] |
T. Li and Y. M. Chen, "Global Classical Solutions for Nonlinear Evolution Equations," Pitman Monogr. Surv. Pure Appl. Math., vol. 45, Longman Scientific and Technical, Harlow, 1992. |
[5] |
T. P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math, 50 (1997), 1113-1182.
doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.3.CO;2-8. |
[6] |
T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Comm. Math. Phys., 169 (1998), 145-173.
doi: 10.1007/s002200050418. |
[7] |
T. P. Liu and Y. Zeng, Large time behavior of solutions general quasilinear hyperbolic-parabolic systems of conservation laws, A. M. S. memoirs, 599 (1997). |
[8] |
Y. Q. Liu, The point-wise estimates of solutions for semi-linear dissipative wave equation, Comm. Pure Appl. Anal., 12 (2013), 237-252.
doi: 10.3934/cpaa.2013.12.237. |
[9] |
Y. Q. Liu and W. K. Wang, The pointwise estimates of solutions for dissipative wave equation in multi-dimensions, Discrete Contin. Dyn. Syst., 20 (2008), 1013-1028.
doi: 10.3934/dcds.2008.20.1013. |
[10] |
M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semiliear dissipative wave equation, Math. Z., 214 (1993), 325-241.
doi: 10.1007/BF02572407. |
[11] |
K. Nishihara, Global asymptotics for the damped wace equation with absotption in higher dimensional space, J. Math. Soc., Japan, 58 (2006), 805-836. |
[12] |
K. Nishihara and H. J. Zhao, Dacay properties of solutions to the Cauchy problem for the damped wace equation with absorption, J. Math. Anal. Appl., 313 (2006), 698-610.
doi: 10.1016/j.jmaa.2005.08.059. |
[13] |
K. Ono, Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations, Discrete Contin. Dyn. Syst., 9 (2003), 651-662.
doi: 10.3934/dcds.2003.9.651. |
[14] |
R. Ikehata, A remark on a critical exponent for the semilinear dissipative wave equation in the one dimensional half space, Differential Integral Equations, 16 (2003), 727-736. |
[15] |
R. Ikehata, K. Nishihara and H. J. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differ. Equation, 226 (2006), 1-29.
doi: 10.1016/j.jde.2006.01.002. |
[16] |
S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (1995), 617-653.
doi: 10.2969/jmsj/04740617. |
[17] |
Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term, Adv. Math. Sci. Appl., 18, (2008), 329-343. |
[18] |
Y. Ueda, T. Nakamura and S. Kawashima, Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space, Kinet. Relat. Models, 1 (2008), 49-64.
doi: 10.3934/krm.2008.1.49. |
[19] |
W. K. Wang and W. J. Wang, The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions, J. Math. Anal. Appl., 366 (2010), 226-241.
doi: 10.1016/j.jmaa.2009.12.013. |
[20] |
W. K. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differ. Equations, 173 (2001), 410-450.
doi: 10.1006/jdeq.2000.3937. |
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