This paper is devoted to an initial-boundary value problem for a class of $ p $-Kirchhoff type parabolic equations. Firstly, we consider this problem with a general nonlocal coefficient $ M(\|\nabla u\|_p^p) $ and a general nonlinearity $ k(t)f(u) $. A new finite time blow-up criterion is established, also, the upper and lower bounds for the blow-up time are derived. Secondly, we deal with the case that $ M(\|\nabla u\|_p^p) = a+b\|\nabla u\|_p^p $, $ k(t)\equiv1 $ and $ f(u) = |u|^{q-1}u $, which was considered by Li and Han [Math. Model. Anal. 2019; 24: 195-217] only for $ q > 2p-1 $. The threshold results for the existence of global and finite time blow-up solutions to this problem are obtained for the case $ 1 < q\leq 2p-1 $, which, together with the results given by Li and Han, shows that $ q = 2p-1 $ is critical for the existence of finite time blow-up solutions to this problem. These results partially generalize and extend some recent ones in previous literature.
Citation: Hui Yang, Futao Ma, Wenjie Gao, Yuzhu Han. Blow-up properties of solutions to a class of $ p $-Kirchhoff evolution equations[J]. Electronic Research Archive, 2022, 30(7): 2663-2680. doi: 10.3934/era.2022136
This paper is devoted to an initial-boundary value problem for a class of $ p $-Kirchhoff type parabolic equations. Firstly, we consider this problem with a general nonlocal coefficient $ M(\|\nabla u\|_p^p) $ and a general nonlinearity $ k(t)f(u) $. A new finite time blow-up criterion is established, also, the upper and lower bounds for the blow-up time are derived. Secondly, we deal with the case that $ M(\|\nabla u\|_p^p) = a+b\|\nabla u\|_p^p $, $ k(t)\equiv1 $ and $ f(u) = |u|^{q-1}u $, which was considered by Li and Han [Math. Model. Anal. 2019; 24: 195-217] only for $ q > 2p-1 $. The threshold results for the existence of global and finite time blow-up solutions to this problem are obtained for the case $ 1 < q\leq 2p-1 $, which, together with the results given by Li and Han, shows that $ q = 2p-1 $ is critical for the existence of finite time blow-up solutions to this problem. These results partially generalize and extend some recent ones in previous literature.
| [1] |
J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland: North-Holland Math. Stud., 30 (1978), 284–346. https://doi.org/10.1016/S0304-0208(08)70870-3 doi: 10.1016/S0304-0208(08)70870-3
|
| [2] |
H. Chen, M. M. Fall, B. Zhang, On isolated singularities of Kirchhoff equations, Adv. Nonlinear Anal., 10 (2021), 102–120. https://doi.org/10.1515/anona-2020-0103 doi: 10.1515/anona-2020-0103
|
| [3] |
W. He, D. Qin, Q. Wu, Existence, multiplicity and nonexistence results for Kirchhoff type equations, Adv. Nonlinear Anal., 10 (2021), 616–635. https://doi.org/10.1515/anona-2020-0154 doi: 10.1515/anona-2020-0154
|
| [4] | A. Hamydy, M. Massar, N. Tsouli, Existence of solutions for $p$-Kirchhoff type problems with critical exponent, Electron. J. Differ. Equ., 105 (2011), 1–8. |
| [5] | E. Dibenedetto, Degenerate Parabolic Equations, Springer, New York, 1993. https://doi.org/10.1007/978-1-4612-0895-2 |
| [6] |
H. Ishii, Asymptotic stability and blowing up of solutions of some nonlinear equations, J. Differ. Equ., 26 (1977), 291–319. https://doi.org/10.1016/0022-0396(77)90196-6 doi: 10.1016/0022-0396(77)90196-6
|
| [7] |
M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. Res. Inst. Math. Sci., 8 (1972), 211–229. https://doi.org/10.2977/prims/1195193108 doi: 10.2977/prims/1195193108
|
| [8] |
H. A. Levine, L. E. Payne, Nonexistence of global weak solutions of classes of nonlinear wave and parabolic equations, J. Math. Anal. Appl., 55 (1976), 329–334. https://doi.org/10.1016/0022-247X(76)90163-3 doi: 10.1016/0022-247X(76)90163-3
|
| [9] |
M. Ghisi, M. Gobbino, Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: time-decay estimates, J. Differ. Equ., 245 (2008), 2979–3007. https://doi.org/10.1016/j.jde.2008.04.017 doi: 10.1016/j.jde.2008.04.017
|
| [10] |
Q. Lin, X. Tian, R. Xu, M. Zhang, Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2095–2107. https://doi.org/10.3934/dcdss.2020160 doi: 10.3934/dcdss.2020160
|
| [11] |
N. Pan, P. Pucci, R. Xu, B. Zhang, Degenerate Kirchhoff-type wave problems involving the fractional Laplacian with nonlinear damping and source terms, J. Evol. Equ., 19 (2019), 615–643. https://doi.org/10.1007/s00028-019-00489-6 doi: 10.1007/s00028-019-00489-6
|
| [12] |
X. Wang, Y. Chen, Y. Yang, J. Li, Kirchhoff-type system with linear weak damping and logarithmic nonlinearities, Nonlinear Anal., 188 (2019), 475–499. https://doi.org/10.1016/j.na.2019.06.019 doi: 10.1016/j.na.2019.06.019
|
| [13] | M. Chipot, T. Savitska, Nonlocal $p$-Laplace equations depending on the $L^p$ norm of the Gradient, Adv. Differ. Equ., 19 (2014), 997–1020. |
| [14] |
Y. Han, 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. https://doi.org/10.1016/j.camwa.2018.01.047 doi: 10.1016/j.camwa.2018.01.047
|
| [15] | S. Zheng, M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptotic Anal., 45 (2005), 301–312. |
| [16] |
Y. Fu, M. Xiang, Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent, Appl. Anal., 95 (2016), 524–544. https://doi.org/10.1080/00036811.2015.1022153 doi: 10.1080/00036811.2015.1022153
|
| [17] |
J. Li, Y. Han, Global existence and finite time blow-up of solutions to a nonlocal $p$-Laplace equation, Math. Model. Anal., 24 (2019), 195–217. https://doi.org/10.3846/mma.2019.014 doi: 10.3846/mma.2019.014
|
| [18] |
L. E. Payne, D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equtions, Israel J. Math., 22 (1975), 273–303. https://doi.org/10.1007/BF02761595 doi: 10.1007/BF02761595
|
| [19] | R. Xu, J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013) 2732–2763. https://doi.org/10.1016/j.jfa.2013.03.010 |
| [20] |
H. A. Levine, Some nonexistence and stability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au+F(u)$, Arch. Ration. Mech. Anal., 51 (1973), 371–386. https://doi.org/10.1007/BF00263041 doi: 10.1007/BF00263041
|
| [21] |
M. Liao, W. Gao, Blow-up phenomena for a nonlocal $p$-Laplace equation with Neumann boundary conditions, Arch. Math., 108 (2017), 313–324. https://doi.org/10.1007/s00013-016-0986-z doi: 10.1007/s00013-016-0986-z
|
| [22] | H. Brezis, Functional Analysis, Sobolev spaces and partial differential equations, Springer, New York, 2010. |
| [23] |
G. A. Philippin, V. Proytcheva, Some remarks on the asymptotic behaviour of the solutions of a class of parabolic problems, Math. Methods Appl. Sci., 29 (2006), 297–307. https://doi.org/10.1002/mma.679 doi: 10.1002/mma.679
|
| [24] |
Y. Han, Finite time blowup for a semilinear pseudo-parabolic equation with general nonlinearity, Appl. Math. Lett., 99 (2020), 1–7. https://doi.org/10.1016/j.aml.2019.07.017 doi: 10.1016/j.aml.2019.07.017
|
| [25] | Y. Li, C. Xie, Blow-up for $p$-Laplacian parabolic equations, Electron. J. Differ. Equ., 20 (2003), 1–12. |
Hui Yang, Futao Ma, Wenjie Gao, Yuzhu Han. Blow-up properties of solutions to a class of $ p $-Kirchhoff evolution equations[J]. Electronic Research Archive, 2022, 30(7): 2663-2680. doi: 10.3934/era.2022136
DownLoad: