American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On some semilinear equation in $R^4$ containing a Laplacian term and involving nonlinearity with exponential growth
  • CPAA Home
  • This Issue
  • Next Article
    Homogenization of bending theory for plates; the case of oscillations in the direction of thickness
November  2015, 14(6): 2169-2183. doi: 10.3934/cpaa.2015.14.2169

Blow up threshold for a parabolic type equation involving space integral and variational structure

Baiyu Liu 1, and  Li Ma 2,

1. 

School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing, 100083, China
2. 

Department of Mathematics, Henan Normal University, Xinxiang, 453007, China

Received  November 2014 Revised  May 2015 Published  September 2015

In this paper, we study a parabolic type equation involving space integrals on a bounded smooth domain. First, using the Banach fixed point theorem, we establish the well-posedness in Lebesgue spaces. Then, with the help of Nehari functional, we find the threshold of the initial data such that the solution either exists globally or blows up in finite time.
Keywords: global existence, potential well method, invariant set, Nonlocal parabolic equation, blow up..
Mathematics Subject Classification: Primary: 35Kxx; 35K2.
Citation: Baiyu Liu, Li Ma. Blow up threshold for a parabolic type equation involving space integral and variational structure. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2169-2183. doi: 10.3934/cpaa.2015.14.2169
References:
[1]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford, 28 (1977), 473-486.  Google Scholar

[2]

F. Dickstein, N. Mizoguchi, P. Souplet and F. Weissler, Transversality of stable and Nehari manifolds for a semilinear heat equation, Calc. Var., 42 (2011), 547-562. doi: 10.1007/s00526-011-0397-8.  Google Scholar

[3]

G. Filippo and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Diff. Int. Eq., 18 (2005), 961-990.  Google Scholar

[4]

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

[5]

S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284. doi: 10.1007/s002850000047.  Google Scholar

[6]

H. Hiroki and Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac., 34 (1991), 475-494.  Google Scholar

[7]

R. Ikehata and and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1992), 475-491.  Google Scholar

[8]

R. Ikehata and T. Suzuki, Semilinear parabolic equations involving critical Sobolev exponent: local and asymptotic behavior of solutions, Diff. Int. Eq., 13 (2000), 869-901. Google Scholar

[9]

A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic beating: Part 1: Model derivation and some special cases, European Journal of Applied Mathematics, 6 (1995), 127-144. Google Scholar

[10]

B. Y. Liu and L. Ma, Invariant sets and the blow up threshold for a nonlocal equation of parabolic type, Nonlinear Analysis: Theory, Methods and Applications, 110 (2014), 141-156. doi: 10.1016/j.na.2014.08.004.  Google Scholar

[11]

Y. C. Liu and J. S. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Analysis: Theory, Methods and Applications, 64 (2006), 2665-2687. doi: 10.1016/j.na.2005.09.011.  Google Scholar

[12]

L. Ma, Global existence and blow-up results for a classical semilinear parabolic equation, Chinese Annals of Mathematics, Series B, 34 (2013), 587-592. doi: 10.1007/s11401-013-0778-8.  Google Scholar

[13]

L. Ma, Blow-up for semilinear parabolic equations with critical sobolev exponent, Commun. Pur. Appl. Anal., 12 (2013), 1103-1110. doi: 10.3934/cpaa.2013.12.1103.  Google Scholar

[14]

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

[15]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics, 22 (1975), 273-303.  Google Scholar

[16]

P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenianae, 68 (1999), 195-203.  Google Scholar

[17]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, $1^{st}$ edition, Birkhauser Advanced Text, Basel/Boston/Berlin, 2007. doi: 978-3-7643-8441-8.  Google Scholar

[18]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Archive for Rational Mechanics and Analysis, 30 (1968), 148-172.  Google Scholar

[19]

J. W. J. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I Travelling wavefronts on unbounded domains, J. Amer. Math. Soc., P. Lond. Math. Soc. A, 2001, 457 (2012), 1841-1853. doi: 10.1098/rspa.2001.0789.  Google Scholar

[20]

T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. J., 57 (2008), 3365-3396. doi: 10.1512/iumj.2008.57.3269.  Google Scholar

[21]

R. Z. Xu, Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Quarterly of Applied Mathematics, 68 (2010), 459-468. doi: 10.1090/S0033-569X-2010-01197-0.  Google Scholar

[22]

G. Yoshikazu, A bound for global solutions of semilinear heat equations, Commun. Math. Phys., 103 (1986), 415-421.  Google Scholar

show all references

References:
[1]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford, 28 (1977), 473-486.  Google Scholar

[2]

F. Dickstein, N. Mizoguchi, P. Souplet and F. Weissler, Transversality of stable and Nehari manifolds for a semilinear heat equation, Calc. Var., 42 (2011), 547-562. doi: 10.1007/s00526-011-0397-8.  Google Scholar

[3]

G. Filippo and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Diff. Int. Eq., 18 (2005), 961-990.  Google Scholar

[4]

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

[5]

S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284. doi: 10.1007/s002850000047.  Google Scholar

[6]

H. Hiroki and Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac., 34 (1991), 475-494.  Google Scholar

[7]

R. Ikehata and and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1992), 475-491.  Google Scholar

[8]

R. Ikehata and T. Suzuki, Semilinear parabolic equations involving critical Sobolev exponent: local and asymptotic behavior of solutions, Diff. Int. Eq., 13 (2000), 869-901. Google Scholar

[9]

A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic beating: Part 1: Model derivation and some special cases, European Journal of Applied Mathematics, 6 (1995), 127-144. Google Scholar

[10]

B. Y. Liu and L. Ma, Invariant sets and the blow up threshold for a nonlocal equation of parabolic type, Nonlinear Analysis: Theory, Methods and Applications, 110 (2014), 141-156. doi: 10.1016/j.na.2014.08.004.  Google Scholar

[11]

Y. C. Liu and J. S. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Analysis: Theory, Methods and Applications, 64 (2006), 2665-2687. doi: 10.1016/j.na.2005.09.011.  Google Scholar

[12]

L. Ma, Global existence and blow-up results for a classical semilinear parabolic equation, Chinese Annals of Mathematics, Series B, 34 (2013), 587-592. doi: 10.1007/s11401-013-0778-8.  Google Scholar

[13]

L. Ma, Blow-up for semilinear parabolic equations with critical sobolev exponent, Commun. Pur. Appl. Anal., 12 (2013), 1103-1110. doi: 10.3934/cpaa.2013.12.1103.  Google Scholar

[14]

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

[15]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics, 22 (1975), 273-303.  Google Scholar

[16]

P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenianae, 68 (1999), 195-203.  Google Scholar

[17]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, $1^{st}$ edition, Birkhauser Advanced Text, Basel/Boston/Berlin, 2007. doi: 978-3-7643-8441-8.  Google Scholar

[18]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Archive for Rational Mechanics and Analysis, 30 (1968), 148-172.  Google Scholar

[19]

J. W. J. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I Travelling wavefronts on unbounded domains, J. Amer. Math. Soc., P. Lond. Math. Soc. A, 2001, 457 (2012), 1841-1853. doi: 10.1098/rspa.2001.0789.  Google Scholar

[20]

T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. J., 57 (2008), 3365-3396. doi: 10.1512/iumj.2008.57.3269.  Google Scholar

[21]

R. Z. Xu, Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Quarterly of Applied Mathematics, 68 (2010), 459-468. doi: 10.1090/S0033-569X-2010-01197-0.  Google Scholar

[22]

G. Yoshikazu, A bound for global solutions of semilinear heat equations, Commun. Math. Phys., 103 (1986), 415-421.  Google Scholar

[1]

Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042
[2]

Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134
[3]

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
[4]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[5]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072
[6]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111
[7]

Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519
[8]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[9]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[10]

Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535
[11]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051
[12]

Xiaoli Zhu, Fuyi Li, Ting Rong. Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2465-2485. doi: 10.3934/cpaa.2015.14.2465
[13]

Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021121
[14]

Walter A. Strauss, Kimitoshi Tsutaya. Existence and blow up of small amplitude nonlinear waves with a negative potential. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 175-188. doi: 10.3934/dcds.1997.3.175
[15]

Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633
[16]

Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169
[17]

Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617
[18]

Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089
[19]

Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183
[20]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (96)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences