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Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity
School of Mathematics, Changchun Normal University, Changchun, 130032, China |
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].
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.
|
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.
|
[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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