American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems
  • ERA Home
  • This Issue
  • Next Article
    The existence of solutions for a shear thinning compressible non-Newtonian models
March  2020, 28(1): 67-90. doi: 10.3934/era.2020005

Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

Jun Zhou ,

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Jun Zhou

Received  September 2019 Revised  November 2019 Published  March 2020

Fund Project: The author is supported by NSF grant 11201380.

This paper deals with the global existence and blow-up of solutions to a inhomogeneous pseudo-parabolic equation with initial value $ u_0 $ in the Sobolev space $ H_0^1( \Omega) $, where $ \Omega\subset \mathbb{R}^n $ ($ n\geq1 $ is an integer) is a bounded domain. By using the mountain-pass level $ d $ (see (14)), the energy functional $ J $ (see (12)) and Nehari function $ I $ (see (13)), we decompose the space $ H_0^1( \Omega) $ into five parts, and in each part, we show the solutions exist globally or blow up in finite time. Furthermore, we study the decay rates for the global solutions and lifespan (i.e., the upper bound of blow-up time) of the blow-up solutions. Moreover, we give a blow-up result which does not depend on $ d $. By using this theorem, we prove the solution can blow up at arbitrary energy level, i.e. for any $ M\in \mathbb{R} $, there exists $ u_0\in H_0^1( \Omega) $ satisfying $ J(u_0) = M $ such that the corresponding solution blows up in finite time.

Keywords: Inhomogeneous pseudo-parabolic equation, global existence, blow-up, Lifespan, decay estimation, ground-state solution.
Mathematics Subject Classification: Primary: 35K70, 35B05; Secondary: 35B40.
Citation: 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
References:
[1]

G. Barenblat, I. Zheltov and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303.  doi: 10.1016/0021-8928(60)90107-6.  Google Scholar

[2]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47–78. doi: 10.1098/rsta.1972.0032.  Google Scholar

[3]

Y. Cao and J. X. Yin, Small perturbation of a semilinear pseudo-parabolic equation, Discrete Contin. Dyn. Syst., 36 (2016), 631–642. doi: 10.3934/dcds.2016.36.631.  Google Scholar

[4]

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

[5]

Y. Cao, Z. Y. Wang and J. X. Yin., A semilinear pseudo-parabolic equation with initial data non-rarefied at $\infty$, J. Func. Anal., 277 (2019), 3737–3756. doi: 10.1016/j.jfa.2019.05.014.  Google Scholar

[6]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1990 French original by Yvan Martel and revised by the authors.  Google Scholar

[7]

H. F. Di, Y. D. Shang and X. M. Peng, Blow-up phenomena for a pseudo-parabolic equation with variable exponents, Appl. Math. Lett., 64 (2017), 67–73. doi: 10.1016/j.aml.2016.08.013.  Google Scholar

[8]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t} = \Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109–124.  Google Scholar

[9]

Y. Z. Han, Finite time blowup for a semilinear pseudo-parabolic equation with general nonlinearity, Appl. Math. Lett., 99 (2020), 105986, 7pp. doi: 10.1016/j.aml.2019.07.017.  Google Scholar

[10]

S. M. Ji, J. X. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 261 (2016), 5446–5464. doi: 10.1016/j.jde.2016.08.017.  Google Scholar

[11]

H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form $Pu_tt = Au + F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1–21. doi: 10.2307/1996814.  Google Scholar

[12]

Z. P. Li and W. J. Du, Cauchy problems of pseudo-parabolic equations with inhomogeneous terms, Z. Angew. Math. Phys., 66 (2015), 3181–3203. doi: 10.1007/s00033-015-0558-2.  Google Scholar

[13]

W. J. Liu and J. Y. Yu, A note on blow-up of solution for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 274 (2018), 1276–1283. doi: 10.1016/j.jfa.2018.01.005.  Google Scholar

[14]

Y. C. Liu and J. S. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665–2687. doi: 10.1016/j.na.2005.09.011.  Google Scholar

[15]

P. Luo, Blow-up phenomena for a pseudo-parabolic equation, Math. Methods Appl. Sci., 38 (2015), 2636-2641.  doi: 10.1002/mma.3253.  Google Scholar

[16]

M. Marras, S. V.-Piro and G. Viglialoro, Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2291–2300. doi: 10.3934/dcdsb.2017096.  Google Scholar

[17]

V. Padrón, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Tran. Amer. Math. Soc., 356 (2004), 2739–2756. doi: 10.1090/S0002-9947-03-03340-3.  Google Scholar

[18]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273–303. doi: 10.1007/BF02761595.  Google Scholar

[19]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148–172. doi: 10.1007/BF00250942.  Google Scholar

[20]

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

[21]

F. L. Sun, L. S. Liu and Y. H. Wu, Finite time blow-up for a class of parabolic or pseudo-parabolic equations, Comput. Math. Appl., 75 (2018), 3685–3701. doi: 10.1016/j.camwa.2018.02.025.  Google Scholar

[22]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences., Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[23]

T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1–26. doi: 10.1007/BF00250690.  Google Scholar

[24]

G. Y. Xu and J. Zhou, Lifespan for a semilinear pseudo-parabolic equation, Math. Methods Appl. Sci., 41 (2018), 705–713.  Google Scholar

[25]

R. Z. Xu and Y. Niu, Addendum to "Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations" [J. Func. Anal., 264 (2013) 2732–2763] [ MR3045640], J. Funct. Anal., 270 (2016), 4039–4041. doi: 10.1016/j.jfa.2016.02.026.  Google Scholar

[26]

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

[27]

R. Z. Xu, X. C. Wang and Y. B. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83 (2018), 176–181. doi: 10.1016/j.aml.2018.03.033.  Google Scholar

[28]

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

[29]

X. L. Zhu, F. Y. Li and Y. H. Li, Some sharp results about the global existence and blowup of solutions to a class of pseudo-parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 1311–1331. doi: 10.1017/S0308210516000494.  Google Scholar

show all references

References:
[1]

G. Barenblat, I. Zheltov and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303.  doi: 10.1016/0021-8928(60)90107-6.  Google Scholar

[2]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47–78. doi: 10.1098/rsta.1972.0032.  Google Scholar

[3]

Y. Cao and J. X. Yin, Small perturbation of a semilinear pseudo-parabolic equation, Discrete Contin. Dyn. Syst., 36 (2016), 631–642. doi: 10.3934/dcds.2016.36.631.  Google Scholar

[4]

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

[5]

Y. Cao, Z. Y. Wang and J. X. Yin., A semilinear pseudo-parabolic equation with initial data non-rarefied at $\infty$, J. Func. Anal., 277 (2019), 3737–3756. doi: 10.1016/j.jfa.2019.05.014.  Google Scholar

[6]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1990 French original by Yvan Martel and revised by the authors.  Google Scholar

[7]

H. F. Di, Y. D. Shang and X. M. Peng, Blow-up phenomena for a pseudo-parabolic equation with variable exponents, Appl. Math. Lett., 64 (2017), 67–73. doi: 10.1016/j.aml.2016.08.013.  Google Scholar

[8]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t} = \Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109–124.  Google Scholar

[9]

Y. Z. Han, Finite time blowup for a semilinear pseudo-parabolic equation with general nonlinearity, Appl. Math. Lett., 99 (2020), 105986, 7pp. doi: 10.1016/j.aml.2019.07.017.  Google Scholar

[10]

S. M. Ji, J. X. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 261 (2016), 5446–5464. doi: 10.1016/j.jde.2016.08.017.  Google Scholar

[11]

H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form $Pu_tt = Au + F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1–21. doi: 10.2307/1996814.  Google Scholar

[12]

Z. P. Li and W. J. Du, Cauchy problems of pseudo-parabolic equations with inhomogeneous terms, Z. Angew. Math. Phys., 66 (2015), 3181–3203. doi: 10.1007/s00033-015-0558-2.  Google Scholar

[13]

W. J. Liu and J. Y. Yu, A note on blow-up of solution for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 274 (2018), 1276–1283. doi: 10.1016/j.jfa.2018.01.005.  Google Scholar

[14]

Y. C. Liu and J. S. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665–2687. doi: 10.1016/j.na.2005.09.011.  Google Scholar

[15]

P. Luo, Blow-up phenomena for a pseudo-parabolic equation, Math. Methods Appl. Sci., 38 (2015), 2636-2641.  doi: 10.1002/mma.3253.  Google Scholar

[16]

M. Marras, S. V.-Piro and G. Viglialoro, Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2291–2300. doi: 10.3934/dcdsb.2017096.  Google Scholar

[17]

V. Padrón, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Tran. Amer. Math. Soc., 356 (2004), 2739–2756. doi: 10.1090/S0002-9947-03-03340-3.  Google Scholar

[18]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273–303. doi: 10.1007/BF02761595.  Google Scholar

[19]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148–172. doi: 10.1007/BF00250942.  Google Scholar

[20]

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

[21]

F. L. Sun, L. S. Liu and Y. H. Wu, Finite time blow-up for a class of parabolic or pseudo-parabolic equations, Comput. Math. Appl., 75 (2018), 3685–3701. doi: 10.1016/j.camwa.2018.02.025.  Google Scholar

[22]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences., Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[23]

T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1–26. doi: 10.1007/BF00250690.  Google Scholar

[24]

G. Y. Xu and J. Zhou, Lifespan for a semilinear pseudo-parabolic equation, Math. Methods Appl. Sci., 41 (2018), 705–713.  Google Scholar

[25]

R. Z. Xu and Y. Niu, Addendum to "Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations" [J. Func. Anal., 264 (2013) 2732–2763] [ MR3045640], J. Funct. Anal., 270 (2016), 4039–4041. doi: 10.1016/j.jfa.2016.02.026.  Google Scholar

[26]

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

[27]

R. Z. Xu, X. C. Wang and Y. B. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83 (2018), 176–181. doi: 10.1016/j.aml.2018.03.033.  Google Scholar

[28]

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

[29]

X. L. Zhu, F. Y. Li and Y. H. Li, Some sharp results about the global existence and blowup of solutions to a class of pseudo-parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 1311–1331. doi: 10.1017/S0308210516000494.  Google Scholar

Figure 1.  The results for $ J(u_0)\leq d $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The graphs of $ f $ and $ g $
Figure Options

Download full-size image

Download as PowerPoint slide

[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]

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
[5]

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
[6]

Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016
[7]

Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117
[8]

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
[9]

Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

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
[16]

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
[17]

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
[18]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, , () : -. doi: 10.3934/era.2020088
[19]

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
[20]

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

2018 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (185)
  • HTML views (383)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences