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Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term
1. | School of Mathematics, Jilin University, Changchun 130012, China, China |
References:
[1] |
S. N. Antontsev and S. I. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Pub. Math., 53 (2009), 355-399.
doi: 10.5565/PUBLMAT_53209_04. |
[2] |
S. N. Antontsev and S. I. Shmarev, Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math., 234 (2010), 2633-2645.
doi: 10.1016/j.cam.2010.01.026. |
[3] |
G. Akagi and K. Matsuura, Nonlinear diffusion equations driven by the $p(.)-$Laplacian, Nonlinear Differ. Equ. Appl., 20 (2013), 37-64.
doi: 10.1007/s00030-012-0153-6. |
[4] |
C. Budd, B. Dold and A. Stuart, Blowup in a partial differential equation with conserved first integral, SIAM J. Appl. Math., 53 (1993), 718-742.
doi: 10.1137/0153036. |
[5] |
Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
[6] |
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[7] |
L. Diening, P. Harjulehto, P. Hästö and M. Rûžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg, 2011.
doi: 10.1007/978-3-642-18363-8. |
[8] |
X. L. Fan and Q. H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. TMA., 52 (2003), 1843-1852.
doi: 10.1016/S0362-546X(02)00150-5. |
[9] |
R. Ferreira, A. de Pablo, M. Pérez-Llanos and J. D. Rossi, Critical exponents for a semilinear parabolic equation with variable reaction, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1027-1042.
doi: 10.1017/S0308210510000399. |
[10] |
B. Guo and W. J. Gao, Study of weak solutions for parabolic equations with nonstandard growth conditions, J. Math. Anal. Appl., 374 (2011), 374-384.
doi: 10.1016/j.jmaa.2010.09.039. |
[11] |
W. J. Gao and Y. Z. Han, Blow-up of a nonlocal semilinear parabolic equation with positive initial energy, Appl. Math. Lett., 24 (2011), 784-788.
doi: 10.1016/j.aml.2010.12.040. |
[12] |
B. Guo and W. J. Gao, Existence and localization of weak solutions of nonlinear parabolic equations with variable exponent of nonlinearity, Ann. Math Pura Appl., 191 (2012), 551-562.
doi: 10.1007/s10231-011-0196-z. |
[13] |
B. Guo and W. J. Gao, Non-extinction of Solutions to a fast diffusive p-Laplace equation with Neumann boundary conditions, J. Math. Anal. Appl., 422 (2015), 1527-1531.
doi: 10.1016/j.jmaa.2014.09.006. |
[14] |
B. Hu and H. M. Yin, Semilinear parabolic equations with prescribed energy, Rendiconti del Circolo Matematico di Palermo, 44 (1995), 479-505.
doi: 10.1007/BF02844682. |
[15] |
A. S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations, Russian. Math. Surveys., 42 (1987), 169-222. |
[16] |
W. J. Liu and M. X. Wang, Blow-up of solutions for a $p$-Laplacian equation with positive initial energy, Acta Appl. Math., 103 (2008), 141-146.
doi: 10.1007/s10440-008-9225-3. |
[17] |
F. C. Li and C. H. X, Global and blow-up solutions to a p-Laplacian equation with nonlocal source, Compu. Math. Appl., 46 (2003), 1525-1533.
doi: 10.1016/S0898-1221(03)90188-X. |
[18] |
C. Y. Qu, X. L. Bai and S. N. Zheng, Blow-up versus extinction in a nonlocal p-Laplace equation with Neumann boundary conditions, J. Math. Anal. Appl., 412 (2014), 326-333.
doi: 10.1016/j.jmaa.2013.10.040. |
[19] |
S. Z. Lian, W. J. Gao, H. J. Yuan and C. L. Cao, Existence of solutions to initial Dirichlet problem of evolution p(x)-Laplace Equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 29 (2012), 377-399.
doi: 10.1016/j.anihpc.2012.01.001. |
[20] |
M. Ruzicka, Electrorheological Fluids: Modelling and Mathematical Theory, Lecture Notes in Math. 1748. Springer, Berlin. 2000.
doi: 10.1007/BFb0104029. |
[21] |
A. El Soufi, M. Jazar and R. Monneau, A Gamma-convergence arguement for the blow-up of a non-local semilear parabolic equation with Neumann boundary condtions, Ann. Inst. H. Poincare Anal. Non Lineaire, 24 (2007), 17-39.
doi: 10.1016/j.anihpc.2005.09.005. |
[22] |
R. Teman, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[23] |
X. L. Wu, B. Guo and W. J. Gao, Blow-up of solutions for a semilinear parabolic equation involving variable exponent source and positive initial energy, Appl. Math. Lett., 26 (2013), 539-543.
doi: 10.1016/j.aml.2012.12.017. |
[24] |
J. X. Yin and C. H. Jin, Critical extinction and blow-up exponents for fast diffusive p-Laplacian with sources, Math. Methods Appl. Sci., 30 (2007), 1147-1167.
doi: 10.1002/mma.833. |
show all references
References:
[1] |
S. N. Antontsev and S. I. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Pub. Math., 53 (2009), 355-399.
doi: 10.5565/PUBLMAT_53209_04. |
[2] |
S. N. Antontsev and S. I. Shmarev, Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math., 234 (2010), 2633-2645.
doi: 10.1016/j.cam.2010.01.026. |
[3] |
G. Akagi and K. Matsuura, Nonlinear diffusion equations driven by the $p(.)-$Laplacian, Nonlinear Differ. Equ. Appl., 20 (2013), 37-64.
doi: 10.1007/s00030-012-0153-6. |
[4] |
C. Budd, B. Dold and A. Stuart, Blowup in a partial differential equation with conserved first integral, SIAM J. Appl. Math., 53 (1993), 718-742.
doi: 10.1137/0153036. |
[5] |
Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
[6] |
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[7] |
L. Diening, P. Harjulehto, P. Hästö and M. Rûžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg, 2011.
doi: 10.1007/978-3-642-18363-8. |
[8] |
X. L. Fan and Q. H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. TMA., 52 (2003), 1843-1852.
doi: 10.1016/S0362-546X(02)00150-5. |
[9] |
R. Ferreira, A. de Pablo, M. Pérez-Llanos and J. D. Rossi, Critical exponents for a semilinear parabolic equation with variable reaction, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1027-1042.
doi: 10.1017/S0308210510000399. |
[10] |
B. Guo and W. J. Gao, Study of weak solutions for parabolic equations with nonstandard growth conditions, J. Math. Anal. Appl., 374 (2011), 374-384.
doi: 10.1016/j.jmaa.2010.09.039. |
[11] |
W. J. Gao and Y. Z. Han, Blow-up of a nonlocal semilinear parabolic equation with positive initial energy, Appl. Math. Lett., 24 (2011), 784-788.
doi: 10.1016/j.aml.2010.12.040. |
[12] |
B. Guo and W. J. Gao, Existence and localization of weak solutions of nonlinear parabolic equations with variable exponent of nonlinearity, Ann. Math Pura Appl., 191 (2012), 551-562.
doi: 10.1007/s10231-011-0196-z. |
[13] |
B. Guo and W. J. Gao, Non-extinction of Solutions to a fast diffusive p-Laplace equation with Neumann boundary conditions, J. Math. Anal. Appl., 422 (2015), 1527-1531.
doi: 10.1016/j.jmaa.2014.09.006. |
[14] |
B. Hu and H. M. Yin, Semilinear parabolic equations with prescribed energy, Rendiconti del Circolo Matematico di Palermo, 44 (1995), 479-505.
doi: 10.1007/BF02844682. |
[15] |
A. S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations, Russian. Math. Surveys., 42 (1987), 169-222. |
[16] |
W. J. Liu and M. X. Wang, Blow-up of solutions for a $p$-Laplacian equation with positive initial energy, Acta Appl. Math., 103 (2008), 141-146.
doi: 10.1007/s10440-008-9225-3. |
[17] |
F. C. Li and C. H. X, Global and blow-up solutions to a p-Laplacian equation with nonlocal source, Compu. Math. Appl., 46 (2003), 1525-1533.
doi: 10.1016/S0898-1221(03)90188-X. |
[18] |
C. Y. Qu, X. L. Bai and S. N. Zheng, Blow-up versus extinction in a nonlocal p-Laplace equation with Neumann boundary conditions, J. Math. Anal. Appl., 412 (2014), 326-333.
doi: 10.1016/j.jmaa.2013.10.040. |
[19] |
S. Z. Lian, W. J. Gao, H. J. Yuan and C. L. Cao, Existence of solutions to initial Dirichlet problem of evolution p(x)-Laplace Equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 29 (2012), 377-399.
doi: 10.1016/j.anihpc.2012.01.001. |
[20] |
M. Ruzicka, Electrorheological Fluids: Modelling and Mathematical Theory, Lecture Notes in Math. 1748. Springer, Berlin. 2000.
doi: 10.1007/BFb0104029. |
[21] |
A. El Soufi, M. Jazar and R. Monneau, A Gamma-convergence arguement for the blow-up of a non-local semilear parabolic equation with Neumann boundary condtions, Ann. Inst. H. Poincare Anal. Non Lineaire, 24 (2007), 17-39.
doi: 10.1016/j.anihpc.2005.09.005. |
[22] |
R. Teman, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[23] |
X. L. Wu, B. Guo and W. J. Gao, Blow-up of solutions for a semilinear parabolic equation involving variable exponent source and positive initial energy, Appl. Math. Lett., 26 (2013), 539-543.
doi: 10.1016/j.aml.2012.12.017. |
[24] |
J. X. Yin and C. H. Jin, Critical extinction and blow-up exponents for fast diffusive p-Laplacian with sources, Math. Methods Appl. Sci., 30 (2007), 1147-1167.
doi: 10.1002/mma.833. |
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