American Institute of Mathematical Sciences

December  2021, 10(4): 723-732. 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  December 2021 Early access  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 and Control Theory, 2021, 10 (4) : 723-732. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [13] S. Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymp. Anal., 45 (2005), 301-312.

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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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [13] S. Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymp. Anal., 45 (2005), 301-312.
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