American Institute of Mathematical Sciences

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February  2016, 36(2): 631-642. doi: 10.3934/dcds.2016.36.631

Small perturbation of a semilinear pseudo-parabolic equation

Yang Cao 1, and  Jingxue Yin 2,

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024
2. 

School of Math. Sci., South China Normal Univ., Guangzhou 510631

Received  May 2014 Revised  February 2015 Published  August 2015

This paper is concerned with large time behavior of solutions for the Cauchy problem of a semilinear pseudo-parabolic equation with small perturbation. It is revealed that small perturbation may develop large variation of solutions with the evolution of time, which is similar to that for the standard heat equation with nonlinear sources.
Keywords: small perturbation, blow-up, Pseudo-parabolic equation, global existence., large time behavior.
Mathematics Subject Classification: Primary: 35K70, 35A01; Secondary: 35B4.
Citation: Yang Cao, Jingxue Yin. Small perturbation of a semilinear pseudo-parabolic equation. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 631-642. doi: 10.3934/dcds.2016.36.631
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26. doi: 10.1137/0501001.  Google Scholar

[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.  Google Scholar

[14]

T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26. doi: 10.1137/0501001.  Google Scholar

[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.  Google Scholar

[14]

T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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