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February  2016, 36(2): 715-730. doi: 10.3934/dcds.2016.36.715

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

Received  June 2014 Revised  February 2015 Published  August 2015

The authors of this paper study some singular phenomena (blowing-up or vanishing in finite time) of solutions to the initial homogeneous $\hbox{Neumann}$ problem of a nonlinear diffusion equation involving the $p(x,t)$-Laplace operator and a nonlinear source. The variable exponent $p(x,t)$ leads to the failure of some techniques, such as upper-lower solutions technique and the scaling method etc., in studying the problem; it also leads to the lack of some valuable properties such as the monotonicity of the energy integral etc. The authors construct a suitable control functional, improve the regularity of the approximate solutions and obtain a new energy inequality to prove that the solution of the problem with a positive initial energy blows up in finite time. Furthermore, under some appropriate conditions, the authors study the vanishing property and the extinction rate estimate of the solutions to the problem by establishing some inequalities the solutions satisfy. It is worth pointing out that the results are obtained with the assumption that $p_{t}(x,t)$ is only negative and integrable which is weaker than those the most of the other papers required.
Citation: Bin Guo, Wenjie Gao. 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. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715
References:
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S. N. Antontsev and S. I. Shmarev, Anisotropic parabolic equations with variable nonlinearity,, Pub. Math., 53 (2009), 355. doi: 10.5565/PUBLMAT_53209_04.

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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. 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. doi: 10.1007/s00030-012-0153-6.

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C. Budd, B. Dold and A. Stuart, Blowup in a partial differential equation with conserved first integral,, SIAM J. Appl. Math., 53 (1993), 718. 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. doi: 10.1137/050624522.

[6]

E. DiBenedetto, Degenerate Parabolic Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-0895-2.

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L. Diening, P. Harjulehto, P. Hästö and M. Rûžička, Lebesgue and Sobolev Spaces with Variable Exponents,, Lecture Notes in Mathematics, (2017). 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. 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. 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. 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. 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. 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. 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. 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.

[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. 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. 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. 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. doi: 10.1016/j.anihpc.2012.01.001.

[20]

M. Ruzicka, Electrorheological Fluids: Modelling and Mathematical Theory,, Lecture Notes in Math. 1748. Springer, (1748). 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. doi: 10.1016/j.anihpc.2005.09.005.

[22]

R. Teman, Infinite Dimensional Dynamical Systems in Mechanics and Physics,,, $2^{nd}$ edition, (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. 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. 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. 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. 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. 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. 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. doi: 10.1137/050624522.

[6]

E. DiBenedetto, Degenerate Parabolic Equations,, Springer-Verlag, (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, (2017). 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. 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. 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. 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. 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. 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. 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. 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.

[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. 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. 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. 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. doi: 10.1016/j.anihpc.2012.01.001.

[20]

M. Ruzicka, Electrorheological Fluids: Modelling and Mathematical Theory,, Lecture Notes in Math. 1748. Springer, (1748). 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. doi: 10.1016/j.anihpc.2005.09.005.

[22]

R. Teman, Infinite Dimensional Dynamical Systems in Mechanics and Physics,,, $2^{nd}$ edition, (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. 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. doi: 10.1002/mma.833.

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