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Time periodic solutions to Navier-Stokes-Korteweg system with friction
Small perturbation of a semilinear pseudo-parabolic equation
1. | School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024 |
2. | School of Math. Sci., South China Normal Univ., Guangzhou 510631 |
References:
[1] |
C. Bandle, H. A. Levine and Q. S. Zhang, Critical exponents of Fujita type for inhomogeneous parabolic equations and systems, J. Math. Anal. Appl., 251 (2000), 624-648.
doi: 10.1006/jmaa.2000.7035. |
[2] |
T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A, 272 (1972), 47-78.
doi: 10.1098/rsta.1972.0032. |
[3] |
Y. Cao, J. X. Yin and C. P. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations, 246 (2009), 4568-4590.
doi: 10.1016/j.jde.2009.03.021. |
[4] |
P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627.
doi: 10.1007/BF01594969. |
[5] |
C. J. van Duijn, L. A. Peletier and I. S. Pop, A new class of entropy solutions of the Buckley-Leverett equation, SIAM J. Math. Anal., 39 (2007), 507-536.
doi: 10.1137/05064518X. |
[6] |
A. Hasan, O. M. Aamo and B. Foss, Boundary control for a class of pseudo-parabolic differential equations, Systems & Control Letters, 62 (2013), 63-69.
doi: 10.1016/j.sysconle.2012.10.009. |
[7] |
E. I. Kaikina, P. I. Naumkin and I. A. Shishmarev, The Cauchy problem for a Sobolev-type equation with power like nonlinearity, Izv. Math., 69 (2005), 59-111.
doi: 10.1070/IM2005v069n01ABEH000521. |
[8] |
J. R. King and C. M. Cuesta, Small and waiting-time behavior of a Darcy flow model with a dynamic pressure saturation relation, SIAM J. Appl. Math., 66 (2006), 1482-1511.
doi: 10.1137/040610969. |
[9] |
A. B. Al'shin, M. O. Korpusov and A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equtions, De Gruyter Series in Nonlinear Analysis and Applications 15, Walter de Gruyter & Co., Berlin, 2011.
doi: 10.1515/9783110255294. |
[10] |
A. Mikelic, A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure, J. Differential Equations, 248 (2010), 1561-1577.
doi: 10.1016/j.jde.2009.11.022. |
[11] |
J. Serrin and H. H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.
doi: 10.1007/BF02392645. |
[12] |
R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.
doi: 10.1137/0501001. |
[13] |
A. Terracina, Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem, SIAM J. Math. Anal., 43 (2011), 228-252.
doi: 10.1137/090778833. |
[14] |
T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26. |
[15] |
R. Z. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.
doi: 10.1016/j.jfa.2013.03.010. |
[16] |
C. X. Yang, Y. Cao and S. N. Zheng, Second critical exponent and life span for pseudo-parabolic equation, J. Differential Equations, 253 (2012), 3286-3303.
doi: 10.1016/j.jde.2012.09.001. |
[17] |
X. Z. Zeng, The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms, J. Math. Anal. Appl., 332 (2007), 1408-1424.
doi: 10.1016/j.jmaa.2006.11.034. |
[18] |
X. Z. Zeng, Blow-up results and global existence of positive solutions for the inhomogeneous evolution P-Laplacian equations, Nonlinear Anal., 66 (2007), 1290-1301.
doi: 10.1016/j.na.2006.01.026. |
[19] |
Q. S. Zhang, A new critical phenomenon for semilinear parabolic problems, J. Math. Anal. Appl., 219 (1998), 125-139.
doi: 10.1006/jmaa.1997.5825. |
[20] |
Q. S. Zhang, Blow up and global existence of solutions to an inhomogeneous parabolic system, J. Differential Equations, 147 (1998), 155-183.
doi: 10.1006/jdeq.1998.3448. |
show all references
References:
[1] |
C. Bandle, H. A. Levine and Q. S. Zhang, Critical exponents of Fujita type for inhomogeneous parabolic equations and systems, J. Math. Anal. Appl., 251 (2000), 624-648.
doi: 10.1006/jmaa.2000.7035. |
[2] |
T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A, 272 (1972), 47-78.
doi: 10.1098/rsta.1972.0032. |
[3] |
Y. Cao, J. X. Yin and C. P. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations, 246 (2009), 4568-4590.
doi: 10.1016/j.jde.2009.03.021. |
[4] |
P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627.
doi: 10.1007/BF01594969. |
[5] |
C. J. van Duijn, L. A. Peletier and I. S. Pop, A new class of entropy solutions of the Buckley-Leverett equation, SIAM J. Math. Anal., 39 (2007), 507-536.
doi: 10.1137/05064518X. |
[6] |
A. Hasan, O. M. Aamo and B. Foss, Boundary control for a class of pseudo-parabolic differential equations, Systems & Control Letters, 62 (2013), 63-69.
doi: 10.1016/j.sysconle.2012.10.009. |
[7] |
E. I. Kaikina, P. I. Naumkin and I. A. Shishmarev, The Cauchy problem for a Sobolev-type equation with power like nonlinearity, Izv. Math., 69 (2005), 59-111.
doi: 10.1070/IM2005v069n01ABEH000521. |
[8] |
J. R. King and C. M. Cuesta, Small and waiting-time behavior of a Darcy flow model with a dynamic pressure saturation relation, SIAM J. Appl. Math., 66 (2006), 1482-1511.
doi: 10.1137/040610969. |
[9] |
A. B. Al'shin, M. O. Korpusov and A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equtions, De Gruyter Series in Nonlinear Analysis and Applications 15, Walter de Gruyter & Co., Berlin, 2011.
doi: 10.1515/9783110255294. |
[10] |
A. Mikelic, A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure, J. Differential Equations, 248 (2010), 1561-1577.
doi: 10.1016/j.jde.2009.11.022. |
[11] |
J. Serrin and H. H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.
doi: 10.1007/BF02392645. |
[12] |
R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.
doi: 10.1137/0501001. |
[13] |
A. Terracina, Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem, SIAM J. Math. Anal., 43 (2011), 228-252.
doi: 10.1137/090778833. |
[14] |
T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26. |
[15] |
R. Z. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.
doi: 10.1016/j.jfa.2013.03.010. |
[16] |
C. X. Yang, Y. Cao and S. N. Zheng, Second critical exponent and life span for pseudo-parabolic equation, J. Differential Equations, 253 (2012), 3286-3303.
doi: 10.1016/j.jde.2012.09.001. |
[17] |
X. Z. Zeng, The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms, J. Math. Anal. Appl., 332 (2007), 1408-1424.
doi: 10.1016/j.jmaa.2006.11.034. |
[18] |
X. Z. Zeng, Blow-up results and global existence of positive solutions for the inhomogeneous evolution P-Laplacian equations, Nonlinear Anal., 66 (2007), 1290-1301.
doi: 10.1016/j.na.2006.01.026. |
[19] |
Q. S. Zhang, A new critical phenomenon for semilinear parabolic problems, J. Math. Anal. Appl., 219 (1998), 125-139.
doi: 10.1006/jmaa.1997.5825. |
[20] |
Q. S. Zhang, Blow up and global existence of solutions to an inhomogeneous parabolic system, J. Differential Equations, 147 (1998), 155-183.
doi: 10.1006/jdeq.1998.3448. |
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