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Homogenization of bending theory for plates; the case of oscillations in the direction of thickness
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 |
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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