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March  2020, 28(1): 67-90. doi: 10.3934/era.2020005

## Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

 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.

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. 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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
The results for $J(u_0)\leq d$
The graphs of $f$ and $g$
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