Global existence and finite time blow-up of solutions to a nonlocal p-Laplace equation
Abstract
In this paper a class of nonlocal diffusion equations associated with a p-Laplace operator, usually referred to as p-Kirchhoff equations, are studied. By applying Galerkin’s approximation and the modified potential well method, we obtain a threshold result for the solutions to exist globally or to blow up in finite time for subcritical and critical initial energy. The decay rate of the L2 norm is also obtained for global solutions. When the initial energy is supercritical, an abstract criterion is given for the solutions to exist globally or to blow up in finite time, in terms of two variational numbers. These generalize some recent results obtained in [Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(9):3283–3297, 2018].
Keyword : p-Kirchhof, potential well, global existence, blow up, initial energy
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
M. Chipot and T. Savitska. Nonlocal p-Laplace equations depending on the Lp norm of the gradient. Adv. Differential Equations, 19(11–12):997–1020, 2014.
M. Chipot, V. Valente and G. V. Caffarelli. Remarks on a nonlocal problems involving the Dirichlet energy. Rend. Sem. Math. Univ. Padova, 110(4):199–220, 2003
P. D’Ancona and Y. Shibata. On global solvability of non-linear viscoelastic equation in the analytic category. Math. Methods Appl. Sci., 17(6):477–489, 1994. https://doi.org/10.1002/mma.1670170605
P. D’Ancona and S. Spagnolo. Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math., 108(1):247–262, 1992. https://doi.org/10.1007/BF02100605
Y.Q. Fu and M.Q. Xiang. Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent. Appl. Anal., 95(3):524–544, 2016. https://doi.org/10.1080/00036811.2015.1022153
F. Gazzola and T. Weth. Finite time blow up and global solutions for semilinear parabolic equations with initial data at high energy level. Differential Integral Equations, 18(9):961–990, 2005.
M. Ghisi and M. Gobbino. Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: time-decay estimates. J. Differential Equations, 245(10):2979–3007, 2008. https://doi.org/10.1016/j.jde.2008.04.017
Y.Z. Han. A class of fourth-order parabolic equation with arbitrary initial energy. Nonlinear Anal. Real World Appl., 43:451–466, 2018. https://doi.org/10.1016/j.nonrwa.2018.03.009
Y.Z. Han and Q.W. Li. Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy. Computers Math. Appl., 75(9):3283–3297, 2018. https://doi.org/10.1016/j.camwa.2018.01.047
G. Kirchhoff. Mechanik. Teubner, Leipzig, 1883.
H.A. Levine. Some nonexistence and instability theorems for solutions of formally parabolic equation of the form put = −au + Fu. Arch. Rati. Mech. Anal., 51(5):371–386, 1973. https://doi.org/10.1007/BF00263041
Q.W. Li, W.J. Gao and Y.Z. Han. Global existence blow up and extinction for a class of thin-film equation. Nonlinear Anal., 147:96–109, 2016. https://doi.org/10.1016/j.na.2016.08.021
J.L. Lions. Quelques methods de resolution des problem aux limits nonlinears. Dunod, Paris, 1969.
J.L. Lions. On some questions in boundary value problems of mathematical physics. North-Holland Mathematical Studies, 30:284–346, 1978. https://doi.org/10.1016/S0304-0208(08)70870-3
Y.C. Liu. On potential wells and vacuum isolating of solutions for semilinear wave equations. J. Differential Equations, 192(1):155–169, 2003. https://doi.org/10.1016/S0022-0396(02)00020-7
K. Nishihara. On a global solution of some quasilinear hyperbolic equation. Tokyo J. Math., 7(2):437–459, 1984. https://doi.org/10.3836/tjm/1270151737
N. Pan, B. L. Zhang and J. Cao. Degenerate Kirchhoff-type diffusion problems involving the fractional p-Laplacian. Nonlinear Anal. Real World Appl., 37:56–70, 2017. https://doi.org/10.1016/j.nonrwa.2017.02.004
L.E. Payne and D.H. Sattinger. Saddle points and instability of nonlinear hyperbolic equtions. Israel J. Math., 22(3-4):273–303, 1975. https://doi.org/10.1007/BF02761595
P. Pucci, M. Q. Xiang and B. L. Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete Contin. Dyn. Syst., 37:4035–4051, 2017. https://doi.org/10.3934/dcds.2017171
C.Y. Qu and W.S. Zhou. Blow-up and extinction for a thin-film equation with initial-boundary value conditions. J. Math. Anal. Appl., 436(2):796–809, 2016. https://doi.org/10.1016/j.jmaa.2015.11.075
D.H. Sattinger. On global solution of nonlinear hyperbolic equations. Arch. Rati. Mech. Anal., 30(2):148–172, 1968. https://doi.org/10.1007/BF00250942
M. Q. Xiang, V. D. Rˇadulescu and B. L. Zhang. Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions. Nonlinearity, 31(7):3228–3250, 2018. https://doi.org/10.1088/1361-6544/aaba35
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(12):2732-2763, 2013. https://doi.org/10.1016/j.jfa.2013.03.010