# American Institute of Mathematical Sciences

May  2019, 18(3): 1205-1226. doi: 10.3934/cpaa.2019058

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

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

* Corresponding author

Received  March 2018 Revised  July 2018 Published  November 2018

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

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.

Citation: Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058
##### References:
 [1] 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. Google Scholar [2] D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. Google Scholar [3] 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. Google Scholar [4] 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. Google Scholar [5] 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. Google Scholar [6] 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. Google Scholar [7] 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. Google Scholar [8] 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. Google Scholar [9] 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. Google Scholar [10] P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, (1985), Pitman (Advanced Publishing Program), Boston, MA. Google Scholar [11] 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. Google Scholar [12] 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. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] 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. Google Scholar [16] 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. Google Scholar [17] 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. Google Scholar [18] 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. Google Scholar [19] 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. Google Scholar [20] 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. Google Scholar [21] 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. Google Scholar

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##### References:
 [1] 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. Google Scholar [2] D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. Google Scholar [3] 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. Google Scholar [4] 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. Google Scholar [5] 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. Google Scholar [6] 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. Google Scholar [7] 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. Google Scholar [8] 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. Google Scholar [9] 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. Google Scholar [10] P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, (1985), Pitman (Advanced Publishing Program), Boston, MA. Google Scholar [11] 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. Google Scholar [12] 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. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] 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. Google Scholar [16] 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. Google Scholar [17] 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. Google Scholar [18] 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. Google Scholar [19] 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. Google Scholar [20] 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. Google Scholar [21] 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. Google Scholar
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