June  2020, 19(6): 3093-3112. doi: 10.3934/cpaa.2020134

Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity

1. 

School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

2. 

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

* Corresponding author

Received  June 2019 Revised  November 2019 Published  March 2020

Fund Project: This project is supported by the National Natural Science Foundation of China (No. 11671031 and No. 11971061) and the Fundamental Research Funds for the Central Universities FRF-BR-17-013A

This paper is concerned with the initial boundary value problem of a nonlocal parabolic equation. By establishing the comparison principle and studying the long-time behavior of its flow, we find the criteria for finite time blow-up and global existence of solutions respectively, which in particular includes the results of arbitrarily high energy initial data. We also characterize the asymptotic profile to both solutions vanishing at infinity and blowing up in finite time.

Citation: Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134
References:
[1]

T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes, vol. 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[2]

J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Phys. D, 227 (2007), 142-148.  doi: 10.1016/j.physd.2007.01.004.

[3]

R. Chill and A. Fiorenza, Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations, J. Differ. Equ., 228 (2006), 611-632.  doi: 10.1016/j.jde.2006.02.009.

[4]

C. CortazarM. del Pino and M. Elgueta, The problem of uniqueness of the limit in a semilinear heat equation, Commun. Partial Differ. Equ., 24 (1999), 2147-2172.  doi: 10.1080/03605309908821497.

[5]

F. DicksteinN. MizoguchiP. Souplet and F. Weissler, Transversality of stable and Nehari manifolds for a semilinear heat equation, Calc. Var. Partial Differ. Equ., 42 (2011), 547-562.  doi: 10.1007/s00526-011-0397-8.

[6]

Z. Dong and J. Zhou, Global existence and finite time blow-up for a class of thin-film equation, Z. Angew. Math. Phys., 68 (2017), 17. doi: 10.1007/s00033-017-0835-3.

[7]

C. EscuderoF. Gazzola and I. Peral, Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian, J. Math. Pures Appl., 103 (2015), 924-957.  doi: 10.1016/j.matpur.2014.09.007.

[8]

B. Feng, Sharp threshold of global existence and instability of standing wave for the Schr$\ddot{o}$dinger-Hartree equation with a harmonic potential, Nonlinear Anal. Real World Appl., 31 (2016), 132-145.  doi: 10.1016/j.nonrwa.2016.01.012.

[9]

H. Fujita, On the nonlinear equations $\Delta u + e^u = 0$ and $\partial v/\partial t = \Delta v + e^v$, B. Am. Math. Soc., 75 (1969), 132-136.  doi: 10.1090/S0002-9904-1969-12175-0.

[10]

J. Furter and M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80.  doi: 10.1007/BF00276081.

[11]

F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differ. Integral Equ., 18 (2005), 961-990. 

[12]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin, 1998.

[13]

R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475-491. 

[14]

A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating. Ⅱ. General proof of blow-up and asymptotics of runaway, Eur. J. Appl. Math., 6 (1995), 201-224.  doi: 10.1017/S0956792500001807.

[15]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au+F(u)$, Arch. Ration. Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.

[16]

X. Li and B. Liu, Vaccum 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.

[17]

X. Li and B. Liu, Finite time blow-up and global existence for the nonlocal complex Ginzburg-Landau equation, J. Math. Anal. Appl., 466 (2018), 961-985.  doi: 10.1016/j.jmaa.2018.06.038.

[18]

B. Liu and L. Ma, Invariant sets and the blow up threshold for a nonlocal equation of parabolic type, Nonlinear Anal. -Theor., 110 (2014), 141-156.  doi: 10.1016/j.na.2014.08.004.

[19]

B. Liu and L. Ma, Blow up threshold for a parabolic type equation involving space integral and variational structure, Commun. Pure Appl. Anal., 14 (2015), 2169-2183.  doi: 10.3934/cpaa.2015.14.2169.

[20]

L. Ma, Global existence and blow-up results for a classical semilinear parabolic equation, Chin. Ann. Math. Ser. B, 34 (2013), 587-592.  doi: 10.1007/s11401-013-0778-8.

[21]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pures Appl., 91 (2009), 49-79.  doi: 10.1016/j.matpur.2008.09.003.

[22]

C. Miao and J. Zhang, On global solution to the Klein-Gordon-Hartree equation below energy space, J. Differ. Equ., 250 (2011), 3418-3447.  doi: 10.1016/j.jde.2010.12.010.

[23]

C. Miao and J. Zhang, Energy scattering for a Klein-Gordon equation with a cubic convolution, J. Differ. Equ., 257 (2014), 2178-2224.  doi: 10.1016/j.jde.2014.05.036.

[24]

C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction diffusion equations, J. Differ. Equ., 235 (2007), 219-261.  doi: 10.1016/j.jde.2006.12.010.

[25]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Isr. J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[26]

P. Quittner, Boundedness of trajectories of parabolic equations and stationary solutions via dynamical methods, Differ. Integral Equ., 7 (1994), 1547-1556. 

[27]

P. Quittner, Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math., 29 (2003), 757-799. 

[28]

P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhauser Advanced Text, Basel/Boston/Berlin, 2007.

[29]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942.

[30]

M. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Jpn., 17 (1972), 173-193. 

[31]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.

[32]

J. ZhangX. LiY. Wu and L. Caccetta, Stability of standing waves for the Klein-Gordon-Hartree equation, Appl. Anal., 95 (2016), 1000-1012.  doi: 10.1080/00036811.2015.1047832.

[33]

G. Xu and J. Zhou, Global existence and blow-up of solutions to a class of nonlocal parabolic equations, Comput. Math. Appl., 78 (2019), 979-996.  doi: 10.1016/j.camwa.2019.03.018.

[34]

J. Zhou, Blow-up and lifespan of solutions to a nonlocal parabolic equation at arbitrary initial energy level, Appl. Math. Lett., 78 (2018), 118-125.  doi: 10.1016/j.aml.201B.11.013.

show all references

References:
[1]

T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes, vol. 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[2]

J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Phys. D, 227 (2007), 142-148.  doi: 10.1016/j.physd.2007.01.004.

[3]

R. Chill and A. Fiorenza, Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations, J. Differ. Equ., 228 (2006), 611-632.  doi: 10.1016/j.jde.2006.02.009.

[4]

C. CortazarM. del Pino and M. Elgueta, The problem of uniqueness of the limit in a semilinear heat equation, Commun. Partial Differ. Equ., 24 (1999), 2147-2172.  doi: 10.1080/03605309908821497.

[5]

F. DicksteinN. MizoguchiP. Souplet and F. Weissler, Transversality of stable and Nehari manifolds for a semilinear heat equation, Calc. Var. Partial Differ. Equ., 42 (2011), 547-562.  doi: 10.1007/s00526-011-0397-8.

[6]

Z. Dong and J. Zhou, Global existence and finite time blow-up for a class of thin-film equation, Z. Angew. Math. Phys., 68 (2017), 17. doi: 10.1007/s00033-017-0835-3.

[7]

C. EscuderoF. Gazzola and I. Peral, Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian, J. Math. Pures Appl., 103 (2015), 924-957.  doi: 10.1016/j.matpur.2014.09.007.

[8]

B. Feng, Sharp threshold of global existence and instability of standing wave for the Schr$\ddot{o}$dinger-Hartree equation with a harmonic potential, Nonlinear Anal. Real World Appl., 31 (2016), 132-145.  doi: 10.1016/j.nonrwa.2016.01.012.

[9]

H. Fujita, On the nonlinear equations $\Delta u + e^u = 0$ and $\partial v/\partial t = \Delta v + e^v$, B. Am. Math. Soc., 75 (1969), 132-136.  doi: 10.1090/S0002-9904-1969-12175-0.

[10]

J. Furter and M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80.  doi: 10.1007/BF00276081.

[11]

F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differ. Integral Equ., 18 (2005), 961-990. 

[12]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin, 1998.

[13]

R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475-491. 

[14]

A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating. Ⅱ. General proof of blow-up and asymptotics of runaway, Eur. J. Appl. Math., 6 (1995), 201-224.  doi: 10.1017/S0956792500001807.

[15]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au+F(u)$, Arch. Ration. Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.

[16]

X. Li and B. Liu, Vaccum 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.

[17]

X. Li and B. Liu, Finite time blow-up and global existence for the nonlocal complex Ginzburg-Landau equation, J. Math. Anal. Appl., 466 (2018), 961-985.  doi: 10.1016/j.jmaa.2018.06.038.

[18]

B. Liu and L. Ma, Invariant sets and the blow up threshold for a nonlocal equation of parabolic type, Nonlinear Anal. -Theor., 110 (2014), 141-156.  doi: 10.1016/j.na.2014.08.004.

[19]

B. Liu and L. Ma, Blow up threshold for a parabolic type equation involving space integral and variational structure, Commun. Pure Appl. Anal., 14 (2015), 2169-2183.  doi: 10.3934/cpaa.2015.14.2169.

[20]

L. Ma, Global existence and blow-up results for a classical semilinear parabolic equation, Chin. Ann. Math. Ser. B, 34 (2013), 587-592.  doi: 10.1007/s11401-013-0778-8.

[21]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pures Appl., 91 (2009), 49-79.  doi: 10.1016/j.matpur.2008.09.003.

[22]

C. Miao and J. Zhang, On global solution to the Klein-Gordon-Hartree equation below energy space, J. Differ. Equ., 250 (2011), 3418-3447.  doi: 10.1016/j.jde.2010.12.010.

[23]

C. Miao and J. Zhang, Energy scattering for a Klein-Gordon equation with a cubic convolution, J. Differ. Equ., 257 (2014), 2178-2224.  doi: 10.1016/j.jde.2014.05.036.

[24]

C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction diffusion equations, J. Differ. Equ., 235 (2007), 219-261.  doi: 10.1016/j.jde.2006.12.010.

[25]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Isr. J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[26]

P. Quittner, Boundedness of trajectories of parabolic equations and stationary solutions via dynamical methods, Differ. Integral Equ., 7 (1994), 1547-1556. 

[27]

P. Quittner, Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math., 29 (2003), 757-799. 

[28]

P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhauser Advanced Text, Basel/Boston/Berlin, 2007.

[29]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942.

[30]

M. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Jpn., 17 (1972), 173-193. 

[31]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.

[32]

J. ZhangX. LiY. Wu and L. Caccetta, Stability of standing waves for the Klein-Gordon-Hartree equation, Appl. Anal., 95 (2016), 1000-1012.  doi: 10.1080/00036811.2015.1047832.

[33]

G. Xu and J. Zhou, Global existence and blow-up of solutions to a class of nonlocal parabolic equations, Comput. Math. Appl., 78 (2019), 979-996.  doi: 10.1016/j.camwa.2019.03.018.

[34]

J. Zhou, Blow-up and lifespan of solutions to a nonlocal parabolic equation at arbitrary initial energy level, Appl. Math. Lett., 78 (2018), 118-125.  doi: 10.1016/j.aml.201B.11.013.

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