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

Previous Article
The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds
DCDS Home
This Issue
Next Article
Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents

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

Abstract
Related Papers

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.

Keywords: generalized $\hbox{Lebesgue-Sobolev}$ spaces., finite-time blow-up, non-extinction, extinction rates, t)\textrm{-Laplace}$ operator, $p(x.

Mathematics Subject Classification: Primary: 35K55, 35K40; Secondary: 35B6.

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:

[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.

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.

[1]	Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243
[2]	Alfonso C. Casal, Jesús Ildefonso Díaz, José M. Vegas. Finite extinction time property for a delayed linear problem on a manifold without boundary. Conference Publications, 2011, 2011 (Special) : 265-271. doi: 10.3934/proc.2011.2011.265
[3]	Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[4]	Marek Fila, Juan-Luis Vázquez, Michael Winkler. A continuum of extinction rates for the fast diffusion equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1129-1147. doi: 10.3934/cpaa.2011.10.1129
[5]	Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733
[6]	Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103
[7]	José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43
[8]	Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072
[9]	Arno Berger. On finite-time hyperbolicity. Communications on Pure & Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963
[10]	Chao Zhang, Lihe Wang, Shulin Zhou, Yun-Ho Kim. Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2559-2587. doi: 10.3934/cpaa.2014.13.2559
[11]	Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175
[12]	Xia Wang, Shengqiang Liu, Libin Rong. Permanence and extinction of a non-autonomous HIV-1 model with time delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1783-1800. doi: 10.3934/dcdsb.2014.19.1783
[13]	Pablo Álvarez-Caudevilla, V. A. Galaktionov. Blow-up scaling and global behaviour of solutions of the bi-Laplace equation via pencil operators. Communications on Pure & Applied Analysis, 2016, 15 (1) : 261-286. doi: 10.3934/cpaa.2016.15.261
[14]	Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025
[15]	Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[16]	Juan Luis Vázquez. Finite-time blow-down in the evolution of point masses by planar logarithmic diffusion. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 1-35. doi: 10.3934/dcds.2007.19.1
[17]	Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25.
[18]	Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463
[19]	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
[20]	Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences