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On the homogenization of some non-coercive Hamilton--Jacobi--Isaacs equations
The point-wise estimates of solutions for semi-linear dissipative wave equation
| 1. | Department of Mathematics, North China Electric Power University, Beijing 102208 |
References:
| [1] |
V. Belleri and V. Pata, Attractors for semi-linear strongly damped wave equations on $R^3$, Discrete Contin. Dyn. Syst., 7 (2001), 719-735.
doi: 10.3934/dcds.2001.7.719. |
| [2] |
L. C. Evans, "Partial Differential Equations," Graduate Studies in Math., 19, Amer. Math. Soc., Providence, RI, 1998. |
| [3] |
D. Hoff and K. Zumbrun, Point-wise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. angew Math. Phys., 48 (1997), 597-614.
doi: 10.1007/s000330050049. |
| [4] |
T. Hosono and T. Ogawa, Large time behavior and $L^p-L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118.
doi: 10.1016/j.jde.2004.03.034. |
| [5] |
R. Ikehata, K. Nishihara and H. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differential Equations, 226 (2006), 1-29.
doi: 10.1016/j.jde.2006.01.002. |
| [6] |
R. Ikehata and M. Ohta, Critical exponent for semi-linear dissipative wave equation in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 269 (2002), 87-97.
doi: 10.1016/S0022-247X(02)00021-5. |
| [7] |
N. I. Karachalios and N. M. Stavrakakis, Estimates on the dimension of a global attractor for a semi-linear dissipative wave equation on $R^n$, Discrete Contin. Dyn. Syst., 8 (2002), 939-951.
doi: 10.3934/dcds.2002.8.939. |
| [8] |
S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semi-linear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (1995), 617-653.
doi: 10.2969/jmsj/04740617. |
| [9] |
T.-T. Li and Y. Zhou, Breakdown of solutions to $\Box u+u_t=|u|^{1+\alpha}$, Discrete Contin. Dyn. Syst., 1 (1995), 503-520.
doi: 10.3934/dcds.1995.1.503. |
| [10] |
J. Lin, K. Nishihara and J. Zhai, $L^2$-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations, 248 (2010), 403-422.
doi: 10.1016/j.jde.2009.09.022. |
| [11] |
Y. Liu and W. Wang, The point-wise 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. |
| [12] |
M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semi-linear dissipative wave equations, Math. Z., 214 (1993), 325-342.
doi: .10.1007/BF02572407. |
| [13] |
K. Nishihara, Global asymptotics for the damped wave equation with absorption in higher dimensional space, J. Math. Soc. Japan, 58 (2006), 805-836.
doi: 10.2969/jmsj/1156342039. |
| [14] |
K. Nishihara and J. Zhai, Asymptotic behaviors of solutions for time dependent damped wave equations, J. Math. Anal. Appl., 360 (2009), 412-421.
doi: 10.1016/j.jmaa.2009.06.065. |
| [15] |
K. Nishihara and H. Zhao, Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption, J. Math. Anal. Appl., 313 (2006), 598-610.
doi: 10.1016/j.jmaa.2005.08.059. |
| [16] |
K. Ono, Asymptotic behavior of solutions for semi-linear telegraph equations, J. Math. Tokushima Univ., 31 (1997), 11-22. |
| [17] |
K. Ono, Global existence and asymptotic behavior of small solutions for semi-linear dissipative wave equations, Discrete Contin. Dyn. Syst., 9 (2003), 651-662.
doi: 10.3934/dcds.2003.9.651. |
| [18] |
G. Todorova and B. Yordnov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489.
doi: 10.1006/jdeq.2000.3933. |
| [19] |
W. Wang and T. Yang, The point-wise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differential Equations, 173 (2001), 410-450.
doi: 10.1006/jdeq.2000.3937. |
show all references
References:
| [1] |
V. Belleri and V. Pata, Attractors for semi-linear strongly damped wave equations on $R^3$, Discrete Contin. Dyn. Syst., 7 (2001), 719-735.
doi: 10.3934/dcds.2001.7.719. |
| [2] |
L. C. Evans, "Partial Differential Equations," Graduate Studies in Math., 19, Amer. Math. Soc., Providence, RI, 1998. |
| [3] |
D. Hoff and K. Zumbrun, Point-wise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. angew Math. Phys., 48 (1997), 597-614.
doi: 10.1007/s000330050049. |
| [4] |
T. Hosono and T. Ogawa, Large time behavior and $L^p-L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118.
doi: 10.1016/j.jde.2004.03.034. |
| [5] |
R. Ikehata, K. Nishihara and H. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differential Equations, 226 (2006), 1-29.
doi: 10.1016/j.jde.2006.01.002. |
| [6] |
R. Ikehata and M. Ohta, Critical exponent for semi-linear dissipative wave equation in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 269 (2002), 87-97.
doi: 10.1016/S0022-247X(02)00021-5. |
| [7] |
N. I. Karachalios and N. M. Stavrakakis, Estimates on the dimension of a global attractor for a semi-linear dissipative wave equation on $R^n$, Discrete Contin. Dyn. Syst., 8 (2002), 939-951.
doi: 10.3934/dcds.2002.8.939. |
| [8] |
S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semi-linear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (1995), 617-653.
doi: 10.2969/jmsj/04740617. |
| [9] |
T.-T. Li and Y. Zhou, Breakdown of solutions to $\Box u+u_t=|u|^{1+\alpha}$, Discrete Contin. Dyn. Syst., 1 (1995), 503-520.
doi: 10.3934/dcds.1995.1.503. |
| [10] |
J. Lin, K. Nishihara and J. Zhai, $L^2$-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations, 248 (2010), 403-422.
doi: 10.1016/j.jde.2009.09.022. |
| [11] |
Y. Liu and W. Wang, The point-wise 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. |
| [12] |
M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semi-linear dissipative wave equations, Math. Z., 214 (1993), 325-342.
doi: .10.1007/BF02572407. |
| [13] |
K. Nishihara, Global asymptotics for the damped wave equation with absorption in higher dimensional space, J. Math. Soc. Japan, 58 (2006), 805-836.
doi: 10.2969/jmsj/1156342039. |
| [14] |
K. Nishihara and J. Zhai, Asymptotic behaviors of solutions for time dependent damped wave equations, J. Math. Anal. Appl., 360 (2009), 412-421.
doi: 10.1016/j.jmaa.2009.06.065. |
| [15] |
K. Nishihara and H. Zhao, Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption, J. Math. Anal. Appl., 313 (2006), 598-610.
doi: 10.1016/j.jmaa.2005.08.059. |
| [16] |
K. Ono, Asymptotic behavior of solutions for semi-linear telegraph equations, J. Math. Tokushima Univ., 31 (1997), 11-22. |
| [17] |
K. Ono, Global existence and asymptotic behavior of small solutions for semi-linear dissipative wave equations, Discrete Contin. Dyn. Syst., 9 (2003), 651-662.
doi: 10.3934/dcds.2003.9.651. |
| [18] |
G. Todorova and B. Yordnov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489.
doi: 10.1006/jdeq.2000.3933. |
| [19] |
W. Wang and T. Yang, The point-wise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differential Equations, 173 (2001), 410-450.
doi: 10.1006/jdeq.2000.3937. |
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