# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020088

## Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity

 School of Mathematics, Changchun Normal University, Changchun, 130032, China

* Corresponding author: Haixia Li

Received  April 2020 Revised  July 2020 Published  August 2020

Fund Project: The author is supported by NSFC (11626044), by NSF of Changchun Normal University (2015-002) and by Scientific Research Foundation for Talented Scholars of Changchun Normal University (RC2016-008)

In this paper, an initial boundary value problem for a parabolic type Kirchhoff equation with time-dependent nonlinearity is considered. A new blow-up criterion for nonnegative initial energy is given and upper and lower bounds for the blow-up time are also derived. These results partially generalize some recent ones obtained by Han and Li in [Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(2018), 3283-3297].

Citation: Haixia Li. Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity. Evolution Equations & Control Theory, doi: 10.3934/eect.2020088
##### References:
 [1] M. Chipot, V. Valente and G. Vergara Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Sem. Math. Univ. Padova, 110 (2003), 199-220.   Google Scholar [2] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.  doi: 10.1007/BF02100605.  Google Scholar [3] Y. Fu and M. Xiang, Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent, Appl. Anal., 95 (2016), 524-544. doi: 10.1080/00036811.2015.1022153.  Google Scholar [4] Y. Han, A class of fourth-order parabolic equation with arbitrary initial energy, Nonlinear Anal., RWA, 43 (2018), 451-466.  doi: 10.1016/j.nonrwa.2018.03.009.  Google Scholar [5] Y. Han, A new blow-up criterion for non-Newton filtration equations with special medium void, Rocky Mountain J. Math., 48 (2018), 2489-2501.  doi: 10.1216/RMJ-2018-48-8-2489.  Google Scholar [6] Y. Han, W. Gao, Z. Sun and H. Li, Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy, Comput. Math. Appl., 76 (2018), 2477-2483.  doi: 10.1016/j.camwa.2018.08.043.  Google Scholar [7] Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Comput. Math. Appl., 75 (2018), 3283-3297.  doi: 10.1016/j.camwa.2018.01.047.  Google Scholar [8] H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equation of the form $Pu_t=-Au+\mathcal{F}u$, Arch. Ration. Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.  Google Scholar [9] J. Li and Y. Han, Global existence and finite time blow-up of solutions to a nonlocal $p$-Laplace equation, Math. Modelling Anal., 24 (2019), 195-217.  doi: 10.3846/mma.2019.014.  Google Scholar [10] G. A. Philippin, Blow-up phenomena for a class of fourth-order parabolic problems, Proceedings AMS, 143 (2015), 2507-2513.  doi: 10.1090/S0002-9939-2015-12446-X.  Google Scholar [11] C. Qu and W. Zhou, Blow-up and extinction for a thin-film equation with initial-boundary value conditions, J. Math. Anal. Appl., 436 (2016), 796-809.  doi: 10.1016/j.jmaa.2015.11.075.  Google Scholar [12] F. Sun, L. Liu and Y. Wu, Finite time blow-up for a class of parabolic or pseudo-parabolic equations, Computers Math. Appl., 75 (2018), 3685-3701.  doi: 10.1016/j.camwa.2018.02.025.  Google Scholar [13] S. Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymp. Anal., 45 (2005), 301-312.   Google Scholar

show all references

##### References:
 [1] M. Chipot, V. Valente and G. Vergara Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Sem. Math. Univ. Padova, 110 (2003), 199-220.   Google Scholar [2] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.  doi: 10.1007/BF02100605.  Google Scholar [3] Y. Fu and M. Xiang, Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent, Appl. Anal., 95 (2016), 524-544. doi: 10.1080/00036811.2015.1022153.  Google Scholar [4] Y. Han, A class of fourth-order parabolic equation with arbitrary initial energy, Nonlinear Anal., RWA, 43 (2018), 451-466.  doi: 10.1016/j.nonrwa.2018.03.009.  Google Scholar [5] Y. Han, A new blow-up criterion for non-Newton filtration equations with special medium void, Rocky Mountain J. Math., 48 (2018), 2489-2501.  doi: 10.1216/RMJ-2018-48-8-2489.  Google Scholar [6] Y. Han, W. Gao, Z. Sun and H. Li, Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy, Comput. Math. Appl., 76 (2018), 2477-2483.  doi: 10.1016/j.camwa.2018.08.043.  Google Scholar [7] Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Comput. Math. Appl., 75 (2018), 3283-3297.  doi: 10.1016/j.camwa.2018.01.047.  Google Scholar [8] H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equation of the form $Pu_t=-Au+\mathcal{F}u$, Arch. Ration. Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.  Google Scholar [9] J. Li and Y. Han, Global existence and finite time blow-up of solutions to a nonlocal $p$-Laplace equation, Math. Modelling Anal., 24 (2019), 195-217.  doi: 10.3846/mma.2019.014.  Google Scholar [10] G. A. Philippin, Blow-up phenomena for a class of fourth-order parabolic problems, Proceedings AMS, 143 (2015), 2507-2513.  doi: 10.1090/S0002-9939-2015-12446-X.  Google Scholar [11] C. Qu and W. Zhou, Blow-up and extinction for a thin-film equation with initial-boundary value conditions, J. Math. Anal. Appl., 436 (2016), 796-809.  doi: 10.1016/j.jmaa.2015.11.075.  Google Scholar [12] F. Sun, L. Liu and Y. Wu, Finite time blow-up for a class of parabolic or pseudo-parabolic equations, Computers Math. Appl., 75 (2018), 3685-3701.  doi: 10.1016/j.camwa.2018.02.025.  Google Scholar [13] S. Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymp. Anal., 45 (2005), 301-312.   Google Scholar
 [1] 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 [2] Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903 [3] Mohamed Jleli, Bessem Samet. Blow-up for semilinear wave equations with time-dependent damping in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3885-3900. doi: 10.3934/cpaa.2020143 [4] 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 [5] Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141 [6] Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969 [7] Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089 [8] Qiang Lin, Xueteng Tian, Runzhang Xu, Meina Zhang. Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 2095-2107. doi: 10.3934/dcdss.2020160 [9] Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041 [10] Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307 [11] Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705 [12] Jungkwon Kim, Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. On Morawetz estimates with time-dependent weights for the Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6275-6288. doi: 10.3934/dcds.2020279 [13] Chan Liu, Jin Wen, Zhidong Zhang. Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020053 [14] 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 [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] Wenhui Chen, Alessandro Palmieri. A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020085 [17] Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639 [18] Yavar Kian, Alexander Tetlow. Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation. Inverse Problems & Imaging, 2020, 14 (5) : 819-839. doi: 10.3934/ipi.2020038 [19] Mourad Bellassoued, Oumaima Ben Fraj. Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements. Inverse Problems & Imaging, 2020, 14 (5) : 841-865. doi: 10.3934/ipi.2020039 [20] Qingquan Chang, Dandan Li, Chunyou Sun. Random attractors for stochastic time-dependent damped wave equation with critical exponents. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2793-2824. doi: 10.3934/dcdsb.2020033

2019 Impact Factor: 0.953