American Institute of Mathematical Sciences

Advanced
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 - A, 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.  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.  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.  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.  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.  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.  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.  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.  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, (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.  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.  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.  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.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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, (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.  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.  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.  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.  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.   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.  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.  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.  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.  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.  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.  doi: 10.1006/jdeq.1998.3448.  Google Scholar

[1]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051
[2]

Xiaoli Zhu, Fuyi Li, Ting Rong. Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2465-2485. doi: 10.3934/cpaa.2015.14.2465
[3]

Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2291-2300. doi: 10.3934/dcdsb.2017096
[4]

Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535
[5]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[6]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[7]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058
[8]

Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134
[9]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072
[10]

Huafei Di, Yadong Shang, Xiaoxiao Zheng. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 781-801. doi: 10.3934/dcdsb.2016.21.781
[11]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[12]

Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633
[13]

Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042
[14]

Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169
[15]

Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025
[16]

Jun Zhou. Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28 (1) : 67-90. doi: 10.3934/era.2020005
[17]

Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243
[18]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[19]

Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086
[20]

Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences