Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian

Ronghua Jiang; Jun Zhou

doi:10.3934/cpaa.2019058

Article Contents

2019, Volume 18, Issue 3: 1205-1226. Doi: 10.3934/cpaa.2019058

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Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian

Ronghua Jiang^1, and
Jun Zhou^1,2, ,

1.
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China
2.
School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550025, China

* Corresponding author
* Corresponding author

Received: February 28, 2018

Revised: June 30, 2018

Published: May 2019

This work is supported by NSFC grant 11201380 and the Basic and Advanced Research Project of CQC-STC grant cstc2016jcyjA0018

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Abstract

We consider a nonlocal parabolic equation associated with the fractional p-laplace operator, which was studied by Gal and Warm in [On some degenerate non-local parabolic equation associated with the fractional p-Laplacian. Dyn. Partial Differ. Equ., 14(1): 47-77, 2017]. By exploiting the boundary condition and the variational structure of the equation, according to the size of the initial dada, we prove the finite time blow-up, global existence, vacuum isolating phenomenon of the solutions. Furthermore, the upper and lower bounds of the blow-up time for blow-up solutions are also studied. The results generalize the results got by Gal and Warm.

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Mathematics Subject Classification: 35R11, 35K55, 35K65.

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References

	D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, volume 314 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 1996, Springer-Verlag, Berlin. doi: 10.1007/978-3-662-03282-4.
	D. Applebaum , Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004) , 1336-1347.
	M. Caponi and P. Pucci , Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016) , 2099-2129. doi: 10.1007/s10231-016-0555-x.
	H. Chen and S. Y. Tian , Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015) , 4424-4442. doi: 10.1016/j.jde.2015.01.038.
	S. Dipierro , G. Palatucci and E. Valdinoci , Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania), 68 (2013) , 201-216.
	P. Felmer , A. Quaas and J. G. Tan , Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012) , 1237-1262. doi: 10.1017/S0308210511000746.
	A. Fiscella and P. Pucci , p-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017) , 350-378. doi: 10.1016/j.nonrwa.2016.11.004.
	A. Fiscella , R. A Servadei and E. Valdinoci , Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015) , 235-253. doi: 10.5186/aasfm.2015.4009.
	C. G. Gal and M. Warma , On some degenerate non-local parabolic equation associated with the fractional p-Laplacian, Dyn. Partial Differ. Equ., 14 (2017) , 47-77. doi: 10.4310/DPDE.2017.v14.n1.a4.
	P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, (1985), Pitman (Advanced Publishing Program), Boston, MA.
	P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 69 of Classics in Applied Mathematics, 2011, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Reprint of the 1985 original [MR0775683], With a foreword by Susanne C. Brenner. doi: 10.1137/1.9781611972030.ch1.
	H. A. Levine , Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt} = -Au+{\mathcal F}(u)$, Trans. Amer. Math. Soc., 192 (1974) , 1-21. doi: 10.2307/1996814.
	X. L. Li and B. Y. Liu , Vacuum isolating, blow up threshold, and asymptotic behavior of solutions for a nonlocal parabolic equation, J. Math. Phys., 58 (2017) , 101503. doi: 10.1063/1.5004668.
	J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, 1972, Springer-Verlag, New York-Heidelberg, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.
	Y. C. Liu , On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003) , 155-169. doi: 10.1016/S0022-0396(02)00020-7.
	Y. C. Liu and J. S. Zhao , On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006) , 2665-2687. doi: 10.1016/j.na.2005.09.011.
	E. D. Nezza , G. Palatucci and E. Valdinoci , Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012) , 521-573. doi: 10.1016/j.bulsci.2011.12.004.
	F. A. Vaillo, J. M. Mazón, J. D. Rossi and J. J. T. Melero, Nonlocal Diffusion Problems, volume 165 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. doi: 10.1090/surv/165.
	M. Warma , The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015) , 499-547. doi: 10.1007/s11118-014-9443-4.
	M. Q. Xiang , G. M. Bisci , G. H. Tian and B. L. Zhang , Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian, Nonlinearity, 29 (2016) , 357-374. doi: 10.1088/0951-7715/29/2/357.
	R. Z. Xu , Y. B. Yang , B. W. Liu , J. H. Shen and S. B. Huang , Global existence and blowup of solutions for the multidimensional sixth-order "good" Boussinesq equation, Z. Angew. Math. Phys., 66 (2015) , 955-976. doi: 10.1007/s00033-014-0459-9.