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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

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.
Citation: Baiyu Liu, Li Ma. Blow up threshold for a parabolic type equation involving space integral and variational structure. Communications on Pure and 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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[18]

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

[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.

[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.

[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.

[22]

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

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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[18]

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

[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.

[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.

[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.

[22]

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

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